Applicable Mathematics/Matrices

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Matrices[edit]

A matrix is a rectangular array of numbers enclosed in brackets. In a notational sense, what differentiates a list of numbers from a matrix is its format. The numbers are listed so that each number has a certain, specific position between the brackets. Each number, or value, in a matrix is called an entry.

One of the main benefits of matrices is the properties which allow them to be manipulated and used for many different, but useful purposes.

Matrices can vary in size. This variation in size is called dimensions. Just like the dimensions of a room (width x length) matrices have dimensions (number of rows x number of columns). Thus, a 2 x 3 (read 2 by 3) matrix will have 2 rows and 3 columns.

Example of a 2 x 3 matrix:



 M = \begin{bmatrix}
    572 & 98   & 302\\
     1  & 732  & 22\\
  \end{bmatrix}


Another term associated with matrices is address. Like your home address, an address describes where each value, or entry, of a matrix lives. The address is composed of the lowercase letter of the matrix with the row and column number (in that order) as subscripts.


Using the 2 x 3 matrix M as an example, the positions of the values are as follows:

 M = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23}\end{bmatrix} =  \begin{bmatrix} 572 & 98 & 302 & \\ 1 & 732 & 22 \end{bmatrix}
m_{11}= 572  \quad m_{12}= 98  \quad m_{13}= 302  \quad m_{21}= 1  \quad m_{22}= 732  \quad m_{23}= 22

A square matrix is any matrix that has the same number of rows as it does columns.

Example: 2 x 2 or 3 x 3 matrices are both square matrices.

 2 x 2 = \begin{bmatrix} {\color{red}1} & 12\\ 21 & {\color{red}2} \end{bmatrix}

\qquad  3 x 3 = \begin{bmatrix} {\color{red}7} & 9 & 30 \\ 1 & {\color{red}2} & 2 \\ 5 & 43 & {\color{red}6}\end{bmatrix}

Take note of the numbers in red above in the 2 x 2 and 3 x 3 square matrices. These numbers are in the addresses of the main diagonal. The main diagonal of a square matrix is the diagonal from the upper left corner entry to the bottom right corner entry. Notice that only square matrices can have a main diagonal.

Adding and Subtracting Matrices[edit]

To add or subtract matrices, the sum or difference is found when addition or subtraction is applied to corresponding entries.

For example,

 
\begin{bmatrix} {\color{red}7} & {\color{blue}4} \end{bmatrix}  +  \begin{bmatrix}{\color{red}3} & {\color{blue}9} \end{bmatrix} = \begin{bmatrix}{\color{red}10}& {\color{blue}13} \end{bmatrix}

\begin{bmatrix} {\color{red}a_{11}} & {\color{blue}a_{12}} \end{bmatrix} + \begin{bmatrix}{\color{red}b_{11}} & {\color{blue}b_{12}} \end{bmatrix} = \begin{bmatrix}{\color{red}a_{11}+ b_{11}}& {\color{blue}a_{12}+ b_{12}} \end{bmatrix}


 
\begin{bmatrix} {\color{red}7} & {\color{blue}4} \end{bmatrix}  -  \begin{bmatrix}{\color{red}3} & {\color{blue}9} \end{bmatrix} = \begin{bmatrix}{\color{red}4}& {\color{blue}-5} \end{bmatrix}

\begin{bmatrix} {\color{red}a_{11}} & {\color{blue}a_{12}} \end{bmatrix} - \begin{bmatrix}{\color{red}b_{11}} & {\color{blue}b_{12}} \end{bmatrix} = \begin{bmatrix}{\color{red}a_{11}- b_{11}}& {\color{blue}a_{12}- b_{12}} \end{bmatrix}

Since addition or subtraction takes place using corresponding entries, matrices must have the same dimensions in order to complete either operation.

Consider this operation


\begin{bmatrix}
    52\\
    36\\
  \end{bmatrix} +

\begin{bmatrix}
    12 & 16 & 5\\
  \end{bmatrix}
====>
\begin{bmatrix}
    a_{11}\\
    a_{21}\\
  \end{bmatrix} +

\begin{bmatrix}
    b_{11} & b_{12} & b_{13}\\
  \end{bmatrix}\quad 
{\color{red}CAN'T\quad  BE\quad  DONE}

\begin{bmatrix}
     a_{11} + b_{11} & ? + b_{12} & ? + b_{13}\\
     a_{21} + ? & ---        & ---\\
\end{bmatrix}

Now consider these matrices


\begin{bmatrix}
    52 & 4\\
    36 & 18\\
  \end{bmatrix} +

\begin{bmatrix}
    12 & 16\\
    34 & 2\\
  \end{bmatrix}
====>
\begin{bmatrix}
    a_{11} & a_{12}\\
    a_{21} & a_{22}\\
  \end{bmatrix} +

\begin{bmatrix}
    b_{11} & b_{12}\\
    b_{21} & b_{22}
  \end{bmatrix}
=
\begin{bmatrix}
    a_{11}+b_{11}& a_{12}+b_{12}\\
    a_{21}+b_{21} & a_{22}+b_{22}\\
  \end{bmatrix} =
\begin{bmatrix}
    64 & 20\\
    70 & 20\\
  \end{bmatrix}

Properties of Equality for Matrices[edit]

Commutative Property[edit]

Matrix addition is commutative   A + B = B + A


\begin{bmatrix}
    5 & 9 & 0\\
    1 & 3 & 2\\
  \end{bmatrix} +
\begin{bmatrix}
    7 & 8 & 3\\
    4 & 7 & 6\\
  \end{bmatrix}= 
\begin{bmatrix}
    7 & 8 & 3\\
    4 & 7 & 6\\
  \end{bmatrix} +
\begin{bmatrix}
    5 & 9 & 0\\
    1 & 3 & 2\\
  \end{bmatrix}


>______________ 
  \begin{bmatrix}
    12 & 17 & 3\\
     5 & 10 & 8\\
  \end{bmatrix} =
\begin{bmatrix}
    12 & 17 & 3\\
     5 & 10 & 8\\
  \end{bmatrix}

Associative Property[edit]

Matrix addition is associative. A + B + C = (A + B) + C = A + (B + C)

\begin{bmatrix}
    4 & 7\\
    1 & 8\\
  \end{bmatrix} +
\begin{bmatrix}
    3 & 18\\
    24 & 7 \\
  \end{bmatrix} +
\begin{bmatrix}
    1  & 8\\
    14 & 6\\
  \end{bmatrix} =
\Bigg(\begin{bmatrix}
    4 & 7\\
    1 & 8\\
  \end{bmatrix}+

\begin{bmatrix}
    3 & 18\\
    24 & 7 \\
  \end{bmatrix}\Bigg) +

\begin{bmatrix}
    1  & 8\\
    14 & 6\\
  \end{bmatrix} =

\begin{bmatrix}
    4 & 7\\
    1 & 8\\
  \end{bmatrix}+
\Bigg(\begin{bmatrix}
    3 & 18\\
    24 & 7 \\
  \end{bmatrix} +

\begin{bmatrix}
    1  & 8\\
    14 & 6\\
  \end{bmatrix} \Bigg)

\begin{bmatrix}
    4 & 7\\
    1 & 8\\
  \end{bmatrix} +
\begin{bmatrix}
    3 & 18\\
    24 & 7 \\
  \end{bmatrix} +
\begin{bmatrix}
    1  & 8\\
    14 & 6\\
  \end{bmatrix} =
\Bigg(\begin{bmatrix}
    7  & 25\\
    25 & 15\\
  \end{bmatrix}\Bigg) +

\begin{bmatrix}
    1  & 8\\
    14 & 6\\
  \end{bmatrix} \qquad \quad=

\begin{bmatrix}
    4 & 7\\
    1 & 8\\
  \end{bmatrix}+
\Bigg(\begin{bmatrix}
    4  & 26\\
    38 & 13 \\
  \end{bmatrix}\Bigg)

>_________\begin{bmatrix}
    8 & 33\\
   39 & 21\\
  \end{bmatrix} \qquad \quad \quad = \qquad \qquad
\begin{bmatrix}
    8 & 33\\
   39 & 21\\
  \end{bmatrix}  \qquad \qquad \quad
=\begin{bmatrix}
    8 & 33\\
   39 & 21\\
  \end{bmatrix}


Additive Identity[edit]

The zero matrix is the additive identity matrix 'O'.

A Zero matrix is a matrix in which all of the entries are zero. 
O =\begin{bmatrix}
    0 & 0\\
    0 & 0\\
  \end{bmatrix}

Any matrix added to the matrix 'O' will retain its' same values.

Example of additive identity A + O = A


\begin{bmatrix}
    7 & 4 \\
    6 & 9 \\
  \end{bmatrix}
+

\begin{bmatrix}
    0 & 0\\
    0 & 0\\
  \end{bmatrix}

=
\begin{bmatrix}
    7 & 4 \\
    6 & 9 \\
  \end{bmatrix}

Additive Inverse[edit]

It takes two matrices to form a pair of inverses. Two matrices are additive inverses if their sum is the zero matrix. This occurs when the additive inverse of a matrix contains the values opposite of each entry. A + (-A) = O



\begin{bmatrix}
    {\color{red}8} & {\color{red}2} & {\color{red}-1}\\
   {\color{red}-6} & {\color{red}9} & {\color{red}-3}\\
  \end{bmatrix}
+

\begin{bmatrix}
   {\color{blue}-8}& {\color{blue}-2} & {\color{blue}1}\\
   {\color{blue}6} & {\color{blue}-9} & {\color{blue}3}\\
  \end{bmatrix}

=
\begin{bmatrix}
{\color{red}8} + {\color{blue}-8} & {\color{red}2} + {\color{blue}-2} & {\color{red}-1} + {\color{blue}1}\\
   {\color{red}-6} + {\color{blue}6} & {\color{red}9}+{\color{blue}-9} & {\color{red}-3}+{\color{blue}3}\\
  \end{bmatrix}

=

\begin{bmatrix}
    0 & 0 & 0\\
    0 & 0 & 0\\
  \end{bmatrix}


Multiplicative Identity Matrix[edit]

A multiplicative identity matrix, usually denoted by the letter I, is any square matrix that has a value of 1 in all the entries along the main diagonal and 0 in the remaining entries.


I_{2x2} =\begin{bmatrix}
    1 & 0\\
    0 & 1\\
  \end{bmatrix} \qquad
I_{3x3} =\begin{bmatrix}
    1 & 0 & 0\\
    0 & 1 & 0\\
    0 & 0 & 1\\
  \end{bmatrix}

Any matrix multiplied by an identity matrix will retain it's original entries. A x I = A



A = \begin{bmatrix}
6 & 15 & 9\\
23 & 5 & 43\\
\end{bmatrix}
\quad  \quad
I =\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{bmatrix}


See Multiplying Matrices



AI=
\begin{bmatrix}
(6)(1) + (15)(0) + (9)(0) & (6)(0) + (15)(1) + (9)(0) & (6)(0) + (15)(0) + (9)(1)\\
(23)(1) + (5)(0) + (43)(0) & (23)(0) + (5)(1) + (43)(0) & (23)(0) + (5)(0) + (43)(1)\\
\end{bmatrix}



AI = \begin{bmatrix}
6 & 15 & 9\\
23 & 5 & 43\\
\end{bmatrix}

Multiplicative Inverse Matrix[edit]

If matrices  A  x  B = I where I is the identity matrix, then A and B are multiplicative inverses of one another.


A = \begin{bmatrix}
-1 & 0 & 2\\
 4 & 1 & -1\\
 2 & 0 & 1\\ \end{bmatrix} \quad 
\quad B = \begin{bmatrix}
-0.2 & 0 & 0.4\\
 1.2 & 1 & -1.4\\
 0.4 & 0 & 0.2\\
\end{bmatrix}



AB=
\begin{bmatrix}
(-1)(-0.2) + (0)(1.2) + (2)(0.4) & (-1)(0) + (0)(1) + (2)(0) & (-1)(0.4) + (0)(-1.4) + (2)(0.2)\\
(4)(-0.2) + (1)(1.2) + (-1)(0.4) & (4)(0) + (1)(1) + (-1)(0)  & (4)(0.4) + (1)(-1.4) + (-1)(0.2) \\
(2)(-0.2) + (0)(1.2) + (1)(0.4) & (2)(0) + (0)(1) + (1)(0)  & (2)(0.4) + (0)(-1.4) + (1)(0.2) \\
\end{bmatrix} 
AB = I = \begin{bmatrix}
    1 & 0 & 0\\
    0 & 1 & 0\\
    0 & 0 & 1\\
  \end{bmatrix}

Thus, matrices A and B are multiplicative inverses of each other.

Multiplying Matrices[edit]

Back to Multiplicative Identity Matrix


Another useful property of matrices is called a scalar. A scalar is a number located outside of a single matrix. To apply the scalar to the matrix, simply multiply each entry of the matrix by the scalar.

For example,


{\color{red}3}\begin{bmatrix}
    3 & 9 & 15\\
    1 & 7.5 & 2\\
  \end{bmatrix} =
\begin{bmatrix}
    ({\color{red}3})3 & ({\color{red}3}) 9 & ({\color{red}3})15\\
    ({\color{red}3})1 & ({\color{red}3})7.5 &({\color{red}3})2\\
  \end{bmatrix} =
\begin{bmatrix}
    9 & 27 & 45\\
    3 & 22.5 & 6\\
  \end{bmatrix}

In order to multiply two matrices together it is necessary to pay attention to their dimensions. Matrices A and J can be multiplied only if the number of columns in A equals the number of rows in J. Also, another hint is that the product of an a x j and a j x b matrix is an a x b matrix. Notice that the number of columns from the first matrix must equal the number of rows from the second matrix (j = j).

First, we'll look at how to test to see if we can multiply matrices.

Consider matrices Q and R.

 
      
Q = 2  \quad x \quad {\color{red}3}\qquad \quad \qquad R = {\color{red}3} \quad x \quad 2\qquad   
{\color{red}3\ columns} = {\color{red}3\ rows}


Since the number of columns of matrix Q equals the rows of matrix R, these matrices can be multiplied. This will produce a 2 x 2 matrix.


Q = \begin{bmatrix}
{\color{red}1} & {\color{red}7} & {\color{red}9}\\
3 & 5 & 1\\
\end{bmatrix}
\qquad 
x
\qquad 
R = \begin{bmatrix}
{\color{red}8} & 9 \\
{\color{red}3} & 5 \\
{\color{red}4} & 6 \\
\end{bmatrix}

Multiplying Q and R.

During this process, you may find your fingers and mental addition extremely helpful. You can use your left pointer finger to follow the entries in the rows of matrix Q and your right pointer finger to follow the columns of matrix R. To multiply matrices, add the products of consecutive entries in corresponding rows of matrix Q and columns of matrix R.

 
QR=\begin{bmatrix} {\color{red}q_{11}} & {\color{red}q_{12}} &  {\color{red}q_{13}}\\
{\color{red}q_{21}} & {\color{red}q_{22}} &  {\color{red}q_{23}}\\
\end{bmatrix}
\begin{bmatrix} {\color{blue}r_{11}} & {\color{blue}r_{12}} &  \\
{\color{blue}r_{21}} & {\color{blue}r_{22}} \\
{\color{blue}r_{31}} & {\color{blue}r_{32}} \\
\end{bmatrix}=

\begin{bmatrix}
QR_{11}={\color{red}q_{11}} \bullet {\color{blue}r_{11}} \quad  + &  {\color{red}q_{12}}  \bullet  {\color{blue}r_{21}}  \quad  + &  {\color{red}q_{13}} \bullet  {\color{blue}r_{31}}\quad   QR_{12}={\color{red}q_{11}} \bullet {\color{blue}r_{12}} \quad  + &  {\color{red}q_{12}}  \bullet  {\color{blue}r_{22}}  \quad  + &  {\color{red}q_{13}} \bullet  {\color{blue}r_{32}} \\
QR_{21}={\color{red}q_{21}} \bullet {\color{blue}r_{11}} \quad  + &  {\color{red}q_{22}}  \bullet  {\color{blue}r_{21}}  \quad  + &  {\color{red}q_{23}} \bullet  {\color{blue}r_{31}}\quad   QR_{22}={\color{red}q_{21}} \bullet {\color{blue}r_{12}} \quad  + &  {\color{red}q_{22}}  \bullet  {\color{blue}r_{22}}  \quad  + &  {\color{red}q_{23}} \bullet  {\color{blue}r_{32}} \\
\end{bmatrix}



QR = \begin{bmatrix}
1 & 7 & 9\\
3 & 5 & 1\\
\end{bmatrix}

\begin{bmatrix}
8 & 9 \\
3 & 5 \\
4 & 6 \\
\end{bmatrix}
=
\begin{bmatrix}
(1)(8) + (7)(3) + (9)(4) & (1)(9) + (7)(5) + (9)(6)\\
(3)(8) + (5)(3) + (1)(4) & (3)(9) + (5)(5) + (1)(6)\\
\end{bmatrix}

=
\begin{bmatrix}
65 &  98 \\
43 &  58 \\
\end{bmatrix}

Determinants[edit]

Every square matrix has a value called a determinant, and only square matrices have defined determinants. The determinant of a 2x2 square matrix is the difference of the products of the diagonals.


The determinant of a 2 x 2 matrix can be found as follows:


det\begin{bmatrix}
    {\color{red}9} & {\color{blue}2}\\
    {\color{blue}7} & {\color{red}8}\\
  \end{bmatrix}=
({\color{red}9})({\color{red}8}) - ({\color{blue}7})({\color{blue}2}) = 58


The "down" diagonal is in red and the "up" diagonal is in blue. The up diagonals are always subtracted from the down diagonals. Matrices that are larger than a 2 x 2 matrix become a little more complicated when finding the determinant but the same rules apply.

    • The "down" diagonal is not necessarily the same as the main diagonal mentioned earlier. The down diagonal happens to be the main diagonal for a 2 x 2 matrix but larger matrices will have multiple down diagonals and only one main diagonal.


Let's find det\begin{bmatrix}
    1 & 6 & 19\\
    8 & 17 & 5\\
    14 & 9 & 3\\
  \end{bmatrix}


When finding the determinant of a 3 x 3 matrix it is helpful to write the first two columns to the right side of the matrix like so,


det\begin{bmatrix}
    1 & 6 & 19\\
    8 & 17 & 5\\
    14 & 9 & 3\\
  \end{bmatrix}
\begin{matrix} 1 & 6 \\ 8 & 17 \\ 14 & 9 \end{matrix}


As shown above in the 2x2 matrix, the numbers are color coded. The blue numbers, once again, indicate they are used in the up diagonals, the red are used in the down diagonals, and those in magenta are used in both.


det\begin{bmatrix}
    {\color{red}1} & {\color{red}6} & {\color{magenta}19}\\
    8 & {\color{magenta}17} & {\color{magenta}5}\\
    {\color{blue}14} & {\color{blue}9} & {\color{magenta}3}\\
  \end{bmatrix}
\begin{matrix} {\color{blue}1} & {\color{blue}6} \\ {\color{magenta}8} & 17 \\ {\color{red}14} & {\color{red}9} \end{matrix} = ({\color{red}1} * {\color{red}17} * {\color{red}3}) + ({\color{red}6} * {\color{red}5} * {\color{red}14}) + ({\color{red}19} * {\color{red}8} * {\color{red}9}) - \Big[({\color{blue}14} * {\color{blue}17} * {\color{blue}19}) + ({\color{blue}9} * {\color{blue}5} * {\color{blue}1}) + ({\color{blue}3} * {\color{blue}8} * {\color{blue}6})\Big]



= ({\color{red}51} + {\color{red}420} + {\color{red}1,368}) - ({\color{blue}4,522} + {\color{blue}45} + {\color{blue}144}) = -2872


Thus, the determinant of the above 3x3 matrix is -2872.

Practice Problems[edit]

Definitions[edit]

Match the following terms with their definitions.


____address>_______________1. Diagonal from the upper left corner entry to the bottom right corner entry

____ determinant of 2x2 matrix>____________2. A rectangular array of numbers enclosed in brackets

____ dimensions>____________3. Variation in size of a matrix

____ main diagonal>__________4. Any matrix that has the same number of rows as it does columns

____ matrix>________________5. Matrix in which all of the entries are zero

____ scalar>________________6. The difference of the products of the diagonals

____ square matrix>__________7. Number located outside of a single matrix which is multiplied by each entry of the matrix

____ zero matrix>____________8. Describes where each value, or entry, of a matrix lives

Definitions Solutions

Properties[edit]

Match the names of the properties with their equation equivalent.


____Additive Identity>______________1. A + B + C = (A + B) + C = A + (B + C)

____ Additive Inverse>______________2. A x I = A

____ Associative Property>__________3.   A + B = B + A

____ Commutative Property>_________4. A + O = A

____ Multiplicative Identity>__________5. A + (-A) = O

____ Multiplicative Inverse>__________6.  A  x  B = I

Properties Solutions

Adding Matrices[edit]

Add the matrices.

1) 
\begin{bmatrix}
    52\\
    36\\
  \end{bmatrix} +

\begin{bmatrix}
    12\\
     5\\
  \end{bmatrix}


2) 
\begin{bmatrix}
    12 & 7  & 17\\
     6 & 91 & 21\\
  \end{bmatrix} +

\begin{bmatrix}
    0 & 14 & 32\\
    5 & 28 & 1 \\
  \end{bmatrix}


3) 
\begin{bmatrix}
     3 & 65\\
     8 & 32\\
  \end{bmatrix} +

\begin{bmatrix}
    9  & 71 & 16\\
    11 & 4  & 10 \\
  \end{bmatrix}


4) 
\begin{bmatrix}
     4  & 56 & 14\\
     15 & 21 & 35\\
  \end{bmatrix} +

\begin{bmatrix}
    87  & 13\\
    5   & 12\\
  \end{bmatrix}


5) 
\begin{bmatrix}
     2 & 34 & 41 & 23 & 14\\
     5 & 69 & 52 & 6  & 61\\
  \end{bmatrix} +

\begin{bmatrix}
    1  & 4 & 5  & 17 & 53\\
    7  & 8 & 99 & 31 & 93\\
  \end{bmatrix}


6) 
\begin{bmatrix}
     19\\
     32\\
     44\\
  \end{bmatrix} +

\begin{bmatrix}
     12\\
     89\\
     42\\
  \end{bmatrix}

Adding Matrices Solutions

Subtracting Matrices[edit]

Subtract the matrices.

1) 
\begin{bmatrix}
    52\\
    36\\
  \end{bmatrix} -

\begin{bmatrix}
    12\\
     5\\
  \end{bmatrix}


2) 
\begin{bmatrix}
    1 & 16 & 43\\
    6 & 21 & 89\\
  \end{bmatrix} -

\begin{bmatrix}
    8 & 2  & 61\\
    9 & 17 & 26\\
  \end{bmatrix}


3) 
\begin{bmatrix}
    65  & 14\\
    4.5 & 43\\
  \end{bmatrix} -

\begin{bmatrix}
    31 & 13\\
     0 & 24\\
  \end{bmatrix}


4) 
\begin{bmatrix}
    52 & 17 & 88\\
     2 & 63 & 16\\
  \end{bmatrix} -

\begin{bmatrix}
    13 & 18\\
     4 & 11\\
  \end{bmatrix}


5) 
\begin{bmatrix}
    3.4 & 7.6 \\
    5.2 & 9.6\\
    1.2 & 8.8
  \end{bmatrix} -

\begin{bmatrix}
    9.8 & 7.6\\
    5.1 & 2.3\\
    6.9 & 4.5\\
  \end{bmatrix}


Subtracting Matrices Solutions


Multiplying Matrices[edit]

Multiply the matrices or by the scalar to find the product.


1) 
\begin{bmatrix}
    4 & 7\\
    2 & 8\\
    3 & 9\\
  \end{bmatrix}

\begin{bmatrix}
    0 & 6 & 9\\
    7 & 2 & 3\\
  \end{bmatrix}


2) 
\begin{bmatrix}
    4 & 7\\
    2 & 8\\
    3 & 9\\
    1 & 6\\
  \end{bmatrix}

\begin{bmatrix}
    0 & 6 & 17 & 9\\
    7 & 2 & 36 & 12\\
  \end{bmatrix}


3) 
\begin{bmatrix}
    14 & 0 & 1\\
     4 & 2 & 9\\
     1 & 6 & 8\\
  \end{bmatrix}

\begin{bmatrix}
    1  & 4 & 5\\
    1  & 2 & 7\\
    12 & 6 & 8\\
  \end{bmatrix}


4)  8
\begin{bmatrix}
    41 & 17 & 54\\
     2 & 8 & 12\\
     4 & 5 &  3\\
  \end{bmatrix}


5) 
\begin{bmatrix}
    4 & 7\\
    2 & 8\\
    3 & 9\\
  \end{bmatrix}

\begin{bmatrix}
    0 & 6\\
    7 & 2\\
    9 & 3\\
  \end{bmatrix}


Multiplying Matrices Solutions

Determinants[edit]

Find the determinant of the following matrices.


1) 
\begin{bmatrix}
    5 & 1\\
    0 & 6\\
      \end{bmatrix}


2) 
\begin{bmatrix}
    1 & 12 & 4\\
    9 &  6 & 3\\
    2 &  2 & 0\\
  \end{bmatrix}


3) 
\begin{bmatrix}
    14 & 4 & 1\\
     1 & 2 & 7\\
    12 & 6 & 3\\
  \end{bmatrix}


4)  
\begin{bmatrix}
     1 & 4\\
     9 & 3 \\
  \end{bmatrix}


5) 
\begin{bmatrix}
    7 & 0\\
    2 & 6\\
  \end{bmatrix}


Determinants Solutions

Solutions[edit]

Definitions Solutions[edit]

Match the following terms with their definitions.


8 address>_______________1. Diagonal from the upper left corner entry to the bottom right corner entry

6 determinant>____________2. A rectangular array of numbers enclosed in brackets

3 dimensions>____________3. Variation in size of a matrix

1 main diagonal>__________4. Any matrix that has the same number of rows as it does columns

2 matrix>________________5. Matrix in which all of the entries are zero

7 scalar>________________6. The difference of the products of the diagonals

4 square matrix>__________7. Number located outside of a single matrix which is multiplied by each entry of the matrix

5 zero matrix>____________8. Describes where each value, or entry, of a matrix lives

Properties Solutions[edit]

Match the names of the properties with their equation equivalent.


4 Additive Identity>______________1. A + B + C = (A + B) + C = A + (B + C)

5 Additive Inverse>______________2. A x I = A

1 Associative Property>__________3.   A + B = B + A

3 Commutative Property>_________4. A + O = A

2 Multiplicative Identity>__________5. A + (-A) = O

6 Multiplicative Inverse>__________6.  A  x  B = I

Adding Matrices Solutions[edit]

Add the matrices.

1) 
\begin{bmatrix}
    64\\
    41\\
  \end{bmatrix}


2) 
\begin{bmatrix}
    12 & 21  & 49\\
    11 & 119 & 22\\
  \end{bmatrix}


3)\quad Cannot be done because the matrices do not have the same dimensions.


4)\quad Cannot be done because the matrices do not have the same dimensions.


5) 
\begin{bmatrix}
     3 & 38 & 46  & 40 & 67\\
    12 & 77 & 151 & 37 & 154\\
  \end{bmatrix}

6) 
\begin{bmatrix}
     31 \\
     121\\
     86\\
  \end{bmatrix}


Subtracting Matrices Solutions[edit]

Subtract the matrices.

1) 
\begin{bmatrix}
    40\\
    31\\
  \end{bmatrix}


2) 
\begin{bmatrix}
    -7 & 14 & -18\\
    -3 &  4 &  63\\
  \end{bmatrix}


3) 
\begin{bmatrix}
    34  & 1\\
    4.5 & 19\\
  \end{bmatrix}


4) Cannot be done because the matrices do not have the same dimensions.


5) 
\begin{bmatrix}
    -6.4 & 0 \\
    0.1  & 7.3\\
    -5.7 & 4.3\\
  \end{bmatrix}


Multiplying Matrices[edit]

Multiply the matrices or by the scalar to find the product.

1) 
\begin{bmatrix}
    49 & 38 & 57\\
    56 & 28 & 42\\
    63 & 36 & 54\\
  \end{bmatrix}


2) 
\begin{bmatrix}
    49 & 38 & 320 & 120\\
    56 & 28 & 322 & 114\\
    63 & 36 & 375 & 135\\
    42 & 18 & 233 & 81\\
  \end{bmatrix}


3) 
\begin{bmatrix}
    26  & 62 & 78\\
    114 & 74 & 106\\
    103 & 64 & 111\\
  \end{bmatrix}


4) 
\begin{bmatrix}
    328 & 136 & 432\\
     16 &  64 & 96\\
     32 &  40 & 24\\
  \end{bmatrix}


5) Cannot be done because the number of columns from the first matrix does not equal the number of rows in the second matrix.

Determinants Solutions[edit]

Find the determinant of the following matrices.


1) 30


2) 90


3) -198


4) -33


5) 42