# MULTIPLICATION OF MATRICES (2x2, 3x2, 3x3, 2x3 MATRIX MULTIPLICATION)

Matrices, like real numbers, can be multiplied.

The rule is this: Two matrices can be multiplied together only if the number of columns in the first matrix is equal to the number of rows in the bother matrix.

Stated simply, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

For example, the matrix product AB can be obtained if the number of columns in matrix A is the same as the number of rows in matrix B.

## Multiplication of 2x2 matrices

When a 2x2 matrix is multiplied by another 2x2 matrix, the number of columns of the first matrix is the same as the number of rows of the second matrix.

To multiply a 2x2 matrix by another 2x2 matrix, use the following formula:

$$\left[{\begin{array}{cc} a & b \\ c & d \\ \end {array} } \right] × \left[{\begin{array}{cc} e & f \\ g & h \\ \end {array} } \right]=\left[{\begin{array}{cc} ae+bg & af+bh \\ ce+dg & cf+dh \\ \end {array} } \right]$$

This means that a 2x2 matrix multiplication will result in a 2x2 matrix.

The next two examples illustrate 2x2 matrix multiplication

#### Example 1

Find the matrix AB given that $A=\left[{\begin{array}{cc} 3 & 4 \\ -5 & 2 \\ \end {array} } \right]$ and $B=\left[{\begin{array}{cc} 2 & 3 \\ 4 & 0 \\ \end {array} } \right]$

Solution:

Remember that

$\left[{\begin{array}{cc} a & b \\ c & d \\ \end {array} } \right] × \left[{\begin{array}{cc} e & f \\ g & h \\ \end {array} } \right]=\left[{\begin{array}{cc} ae+bg & af+bh \\ ce+dg & cf+dh \\ \end {array} } \right]$

Accordingly.

$AB=\left[{\begin{array}{cc} 3(2)+4(4) & 3(3)+4(0) \\ -5(2)+2(4) & -5(3)+2(0) \\ \end {array} } \right]$

$AB=\left[{\begin{array}{cc}22 & 9 \\ -2 & -15 \\ \end {array} } \right]$

#### Example 2

Find the matrix BA given that $A=\left[{\begin{array}{cc} 3 & 4 \\ -5 & 2 \\ \end {array} } \right]$ and $B=\left[{\begin{array}{cc} 2 & 3 \\ 4 & 0 \\ \end {array} } \right]$

Solution:

This time, Matrix B is multiplied by Matrix A.

Note, in matrix multiplication, $AB≠BA$. This means that the multiplication of matrices is not commutative

$BA=\left[{\begin{array}{cc} 2(3)+3(-5) & 2(4)+3(2) \\ 4(3)+0(-5) & 4(4)+0(2) \\ \end {array} } \right]$

$BA=\left[{\begin{array}{cc} -9 & 14 \\ 12 & 16 \\ \end {array} } \right]$

## Multiplication Of 3x3 Matrices

The product of two 3x3 matrices can be obtained using the formula:

$$\left[{\begin{array}{cccc} a & b & c \\ d & e & f \\ g & h & i \end {array} } \right] × \left[{\begin{array}{cccc} j & k & l \\ m & n & o \\ p & q & r \end {array} } \right]$$

$$\left[{\begin{array}{cccc} aj+bm+cp & ak+bn+cq & al + bo+ cr\\ dj+em+fp & dk+en+fq & dl+eo+fr \\ gf+hm+ip & gk+ hn +iq & gl+ho+ir \end {array} } \right]$$

#### Example 3

Determine Matrix AB given that

$A=\left[{\begin{array}{cccc} -4 & 3 & 2 \\ -5 & 6 & 1 \\ -4 & 7 & 0 \end {array} } \right]$ and $B=\left[{\begin{array}{cccc} 3 & -7 & 8 \\ 0 & 2 & 4 \\ 1 & 2 & 3 \end {array} } \right]$

Solution:

Remember that

$$\left[{\begin{array}{cccc} aj+bm+cp & ak+bn+cq & al + bo+ cr\\ dj+em+fp & dk+en+fq & dl+eo+fr \\ gf+hm+ip & gk+ hn +iq & gl+ho+ir \end {array} } \right]$$

Using the same procedure, AB is equal to

$\left[{\begin{array}{cccc} 4(3)+3(0)+2(1) & 4(-7)+3(2)+2(2) & 4(8) + 3(4)+ 2(3) \\ -5(3)+6(0)+1(1) & -5(-7)+6(2)+1(2) & -5(8) + 6(4)+1(3) \\ -4(3)+7(0)+0(1) & -4(-7)+ 7(2) +0(2) & -4(8)+7(4)+0(3) \end {array} } \right]$

$AB=\left[{\begin{array}{cccc} 14 & -18 & 50 \\ -14 & 49 & -13 \\ -12 & 42 & -4 \end {array} }\right]$

#### Example 4

Determine Matrix BA given that

$A=\left[{\begin{array}{cccc} -4 & 3 & 2 \\ -5 & 6 & 1 \\ -4 & 7 & 0 \end {array} } \right]$ and $B=\left[{\begin{array}{cccc} 3 & -7 & 8 \\ 0 & 2 & 4 \\ 1 & 2 & 3 \end {array} } \right]$

Solution:

BA is

$\left[{\begin{array}{cccc} 3(-4)-7(-5)+8(-4) & 3(3)-7(6)+8(7) & 3(2) -7(1)+ 8(0) \\ 0(-4)+2(-5)+4(-4) & 0(3)+2(6)+4(7) & 0(2) + 2(1)+4(0) \\ 1(-4)+2(-5)+3(-4) & 1(3)+ 2(6) +3(7) & 1(2)+2(1)+3(0) \end {array} } \right]$

$BA=\left[{\begin{array}{cccc} -9 & 23 & -1 \\ -26 & 40 & 2 \\ -26 & 36 & 4 \end {array} }\right]$

## Multiplication of 2x3 Matrix to a 3x2 matrix

The multiplication of a 2x3 matrix and 3x2 matrix is exemplified as follows:

$$\left[{\begin{array}{cccc} a & b & c \\ d & e & f \end {array} } \right] × \left[{\begin{array}{cc} g & j \\ h & k \\ i & l \end {array} } \right]$$

The multiplication of a 2x3 matrix by a 3x2 matrix produces a 2 x 2 matrix, as can be seen.

#### Example 5

Given that $A=\left[{\begin{array}{cccc} 2 & 3 & 5 \\ 6 & 3 & 4 \end {array} } \right]$ and $B=\left[{\begin{array}{cc} 3 & 2 \\ 6 & 8 \\ 4 & 1 & \end {array} } \right]$. Determine the Matrix AB

Solution:

Recall that

$$\left[{\begin{array}{cccc} a & b & c \\ d & e & f \end {array} } \right] × \left[{\begin{array}{cc} g & j \\ h & k \\ i & l \end {array} } \right]$$

Accordingly,

$AB=\left[{\begin{array}{cc} 2(3)+3(6)+5(4) & 2(2)+3(8)+5(1)\\ 6(3)+3(6)+4(4) & 6(2)+3(8)+4(1) \end {array} } \right]$

$AB=\left[{\begin{array}{cc} 44 & 33 \\ 52 & 40 \end{array} } \right]$

## Multiplication of 3x2 matrix and a 2x3 matrix

The following is an example of multiplication of a 3x2 and 2x3 matrix:

$$\left[{\begin{array}{cc} a & d \\ b & e \\ c & f \end {array} } \right] \times \left[{\begin{array}{cccc} g & h & i\\ j & k & l \end{array} } \right]$$

$$\left[{\begin{array}{cccc} ag+dj & ah+dk & ai+dl \\ bg+ej & bh+ek & bi+el \\ cg+ej & ch+ek & ci+el \end {array} } \right]$$

As can be seen, the result of multiplying a 3x2 matrix to a 2x3 matrix gives a 3x3 matrix.

#### Example 6

Obtain matrix BA given that $B=\left[{\begin{array}{cc} 3 & 2 \\ 6 & 8 \\ 4 & 1 \end {array} } \right]$ and $A=\left[{\begin{array}{cccc} 2 & 3 & 5 \\ 6 & 3 & 4 \end {array} } \right]$

Solution:

$BA=\left[{\begin{array}{cccc} 3(2)+2(6) & 3(3)+2(3) & 3(5)+2(5) \\ 6(2)+8(6) & 6(3)+8(3) & 6(5)+8(4) \\ 4(2) +1(6) & 4(3) +1(3) & 4(5)+1(4) \end {array} } \right]$

$BA=\left[{\begin{array}{cccc} 18 & 15 & 23 \\ 60 & 42 & 62 \\ 14 & 15 & 24 \end {array} }\right]$

## Multiplication of row matrix and column matrix

A row matrix is a matrix that has only one row. Here is an example

$\left[{\begin{array}{cccc} a & b & c \end {array} } \right]$

A column matrix is a matrix that has only one column. Here's an example

$$\left[{\begin{array}{cc} d \\ e \\ f \end {array} } \right]$$

To multiply a row matrix by a column matrix, use the following formula

$$\left[{\begin{array}{cccc} a & b & c \end {array} } \right] × \left[{\begin{array}{cc} d \\ e \\ f \end {array} } \right]= \left[{\begin{array}{cccc} a(d)+ b(e)+ c(f) \end {array} } \right]$$

The result of the multiplication of a row matrix to a column matrix gives a 1x1, which is also known as a scalar.

#### Example 7

Determine matrix AB given that $A=\left[{\begin{array}{cccc} 3 & 4 & -2 \end {array} } \right]$ and $B=\left[{\begin{array}{cc} 5\\ -1 \\ -4 \end {array} } \right]$

Solution:

$AB=\left[{\begin{array}{cccc} 3(5)+ 4(-1)+ -4(-2) \end {array} } \right]$

$AB=\left[{\begin{array}{cccc} 15-4+8 \end {array} } \right]$

$AB=\left[{\begin{array}{cccc} 19 \end {array} } \right]$

## Multiplication of column matrix and row matrix

The multiplication of a column matrix to a row matrix will give a 3x3 matrix.

This is exemplified as follow

$$\left[{\begin{array}{cc} d \\ e \\ f \end {array} } \right] × \left[{\begin{array}{cccc} a & b & c \end {array} } \right]=\left[{\begin{array}{cccc} d(a) & d(b) & d(c) \\ e(a) & e(b) & e(c) \\ f(a) & f(b) & f(c) \end {array} } \right]$$

#### Example 8

Determine matrix BA given that $A=\left[{\begin{array}{cccc} 3 & 4 & -2 \end {array} } \right]$ and $B=\left[{\begin{array}{cc} 5\\ -1 \\ -4 \end {array} } \right]$

Solution:

In this case, B (a column matrix) is multiplied to A ( a row matrix), using

$$\left[{\begin{array}{cc} d \\ e \\ f \end {array} } \right] × \left[{\begin{array}{cccc} a & b & c \end {array} } \right]=\left[{\begin{array}{cccc} d(a) & d(b) & d(c) \\ e(a) & e(b) & e(c) \\ f(a) & f(b) & f(c) \end{array} } \right]$$

$BA=\left[{\begin{array}{cccc} 5(3) & 5(4) & 5(-2) \\ -1(3) & -1(4) & -1(-2) \\ -4(3) & -4(4) & -4(-2) \end {array} } \right]$

$BA=\left[{\begin{array}{cccc} 15 & 20 & -10 \\ -3 & -4 & 2 \\ -12 & -16 & 8 \end {array} } \right]$

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