5th Hour Honors Algebra 2 (Spring 2012): Section 9.6

9.6: The Algebra Arithmetic of Matrices

In this section, we will discuss the properties of matrices, and what it takes to add, subtract, and multiply them. There are certain conditions for these matrix operations that you must remember.

ADDITION OF MATRICES

First, let's talk about the addition of two or more matrices.

Note: You may only add two ore more matrices together if they have the same dimensions.

In order to add two or more matrices, you must add the corresponding elements of each matrix together.

For example, if

$a=\begin{bmatrix} 3 & -4\\ 6 & 8 \end{bmatrix}$ $and$ $b=\begin{bmatrix} 0 & 5\\ -2 & 1 \end{bmatrix}$ $,$ $a+b=\begin{bmatrix} 3+0 & -4+5\\ 6+(-2) & 8+1 \end{bmatrix}=\begin{bmatrix} 3 &1 \\ 4& 9 \end{bmatrix}$

SUBTRACTION OF MATRICES

The subtraction of matrices is similar to the addition of matrices, except that unlike addition, it is NOT commutative. The two or more matrices still must have the same dimensions. In order to subtract two or more matrices, you must subtract the corresponding elements of each matrix.

For example, if

$a=\begin{bmatrix} 1 & 4\\ 5 &7 \\ 8&9 \end{bmatrix}$ $and$ $b=\begin{bmatrix} -2 & 8\\ 6& -1\\ 0&4 \end{bmatrix}$ $,$ $a-b=\begin{bmatrix} 1-(-2) &4-8 \\ 5-6& 7-(-1)\\ 8-0&9-4 \end{bmatrix}=\begin{bmatrix} 3 &-4 \\ -1&8 \\ 8 &5 \end{bmatrix}$

MULTIPLICATION OF MATRICES

There are two forms of multiplication of matrices. The first is multiplying a matrix by a real number. Like the addition and subtraction of matrices, this operation is quite simple and just involves distributing the number to each element of the matrix. It will look like this:

$8\begin{bmatrix} 5 &3 & 2\\ -1& 6 &8 \\ 3&- 5 &3 \end{bmatrix}=\begin{bmatrix} 8\cdot 5& 8\cdot 3 &8\cdot 2 \\ 8\cdot (-1)& 8\cdot 6 &8\cdot 8 \\ 8\cdot 3 & 8\cdot(-5) & 8\cdot 3 \end{bmatrix}=\begin{bmatrix} 40 &24 &16 \\ -8& 48 & 64\\ 24 &-40 & 24 \end{bmatrix}$

The second form of the multiplication of matrices is a bit more confusing. Although the matrices can have different dimensions, there are still strict conditions that these dimensions must follow.

The best way to figure out if A) the matrices are multipliable and B) what the dimensions of the resulting matrix will be is to write out the dimensions side-by-side like so:

As the graphic shows, the inside numbers MUST match, or else the matrices cannot be multiplied. To find the dimensions of the resulting matrix, use the outside numbers - the number of rows of the first matrix by the number of columns of the second matrix.

Now the process of multiplying the matrices is quite hard to explain in a blog post. The best way to start is two write out your brackets for the resulting matrix and go element by element slowly. To multiply, use the rows of the first matrix and the columns of the second.

If you are given two matrices:

$a=\begin{bmatrix} 1 & 4 & 2\\ 3& -2 &5 \end{bmatrix}$ $and$ $b=\begin{bmatrix} 1 & 2\\ -4& 3\\ 5 & -6 \end{bmatrix}$ ,

Then their dimensions are 2x3 and 3x2. They will multiply, and the resulting matrix will be 2x2.

It will look like this:

$\begin{bmatrix} (1\cdot 1)+(4\cdot (-4))+(2\cdot 5) & (1\cdot 2)+(4\cdot3 )+(2\cdot (-6))\\ (3\cdot 1)+((-2)\cdot (-4))+(5\cdot 5) & (3\cdot 2)+((-2)\cdot3) +(5\cdot (-6))\end{bmatrix}=\begin{bmatrix} -5 &2 \\ 36 & -30 \end{bmatrix}$