Multiplication and Power of Matrices

The multiplications of matrices are presented using examples and questions with solutions.

Multiplication of Rows and Columns Matrices

Let A be a row matrix of order 1 × p with entries a_1j and B be a column matrix of order p × 1 with entries b_j1. The multiplication of matrix A by matrix B is a 1 × 1 matrix defined by:

Example 1
Matrices A and B are defined by
row column example

Find the matrix A B.

Solution

Multiplication of Matrices

We now apply the idea of multiplying a row by a column to multiplying more general matrices. Let A be an m × p matrix and B be an p × n matrix. Let R₁, R₂, ... R_m be the rows of matrix A and C₁, C₂, ... C_n be the columns of column B and write the two matrices as:
column row multiplication

The product of the two matrices A and B is matrix C of order m × n defined by
column row multiplication result

Example 2

Find the product

[\begin{matrix} 2 & - 1 & 0 \\ 1 & 2 & 2 \end{matrix}] \cdot [\begin{matrix} 2 & - 2 \\ 1 & 0 \\ - 1 & 2 \end{matrix}]

Solution
The matrix on the left has 2 rows R₁ and R₂ the matrix on the right has 2 columns C₁ and C₂. Their product is given by:

[\begin{matrix} 2 & - 1 & 0 \\ 1 & 2 & 2 \end{matrix}] \cdot [\begin{matrix} 2 & - 2 \\ 1 & 0 \\ - 1 & 2 \end{matrix}] = [\begin{matrix} R_{1} \\ R_{2} \end{matrix}] \cdot [\begin{matrix} C_{1} & C_{2} \end{matrix}] = [\begin{matrix} R_{1} \cdot C_{1} & R_{1} \cdot C_{2} \\ R_{2} \cdot C_{1} & R_{2} \cdot C_{2} \end{matrix}]

= [\begin{matrix} [\begin{matrix} 2 & - 1 & 0 \end{matrix}] \cdot [\begin{matrix} 2 \\ 1 \\ - 1 \end{matrix}] & [\begin{matrix} 2 & - 1 & 0 \end{matrix}] \cdot [\begin{matrix} - 2 \\ 0 \\ 2 \end{matrix}] \\ [\begin{matrix} 1 & 2 & 2 \end{matrix}] \cdot [\begin{matrix} 2 \\ 1 \\ - 1 \end{matrix}] & [\begin{matrix} 1 & 2 & 2 \end{matrix}] \cdot [\begin{matrix} - 2 \\ 0 \\ 2 \end{matrix}] \end{matrix}] = [\begin{matrix} (2) (2) + (- 1) (1) + (0) (- 1) & (2) (- 2) + (- 2) (0) + (0) (2) \\ (1) (2) + (2) (1) + (2) (- 1) & (1) (- 2) + (2) (0) + (2) (2) \end{matrix}] = [\begin{matrix} 3 & - 4 \\ 2 & 2 \end{matrix}]

Power of a Matrix

The power of a square matrix A is defined as follows:

A^{0} = I

I

the identity matrix

A^{n} = A A . . . . A

(n times) , where n is a positive integer.
If m and n are positive integers, then

A^{m} A^{n} = A^{m + n}

(A^{m})^{n} = A^{m n}

Properties of Matrix Multiplication

The product $A B$ of two matrices $A$ and $B$ is defined if the number of columns of matrix $A$ is equal to the number of rows of matrix $B$ .
In general, the product of two matrices is not commutative: $A B \neq B A$
Matrix multiplication is associative: $(A B) C = A (B C)$ if all the multiplications are defined.
Matrix multiplication is distributive: $A (B + C) = A B + A C$ and $(A + B) C = A C + B C$
Multiplication by an identity matrix $I$ : $A I = I A = A$ , this holds for square matrices of dimension n by n.
For α and β real: $α (A + B) = α A + α B$
For α and β real: $α (β A) = α β (A)$
For α and β real: $(α + β) A = α A + β A$
For α real: $α (A B) = (α A) B = A (α B)$

Questions on Multiplication of Matrices

Part 1
A, B, C, D and E are matrices with the orders
A: 2 × 3 , B: 3 × 5 , C: 5 × 1 , E: 1 × 5
Which of the following are defined?
1. $A B$
2. $A C$
3. $C E$
4. $E C$
5. $(A B) C$
Part 2
A, B, C, D and E are matrices given by: $A = [\begin{matrix} - 1 & 1 & - 2 \\ 0 & - 2 & 1 \end{matrix}], B = [\begin{matrix} - 1 & 2 & 0 \\ 0 & - 3 & 4 \\ - 1 & - 2 & 3 \end{matrix}], C = [\begin{matrix} - 3 & 2 & 9 & - 5 & 7 \end{matrix}] D = [\begin{matrix} - 2 & 6 \\ - 5 & 2 \end{matrix}], E = [\begin{matrix} 3 \\ 5 \\ - 11 \end{matrix}], F = [\begin{matrix} - 1 & 0 & 2 \\ - 2 & - 3 & 4 \\ 1 & 4 & - 3 \end{matrix}]$ Find if possible:
1. $A B$
2. $B C$
3. $A D$
4. $E F$
5. $F E$
Part 3
Find x and y if $[\begin{matrix} x + y & - 2 \\ x - y & 1 \end{matrix}] [\begin{matrix} 2 & - 1 \\ 0 & - 2 \end{matrix}] = [\begin{matrix} 8 & 0 \\ 12 & - 8 \end{matrix}]$
Part 4
Calculate ${([\begin{matrix} 2 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{matrix}])}^{10}$

Solutions to the Above Questions

Part 1
A, B, C, D and E are matrices with the orders
A: 2 × 3 , B: 3 × 5 , C: 5 × 1 , E: 1 × 5
Which of the following are defined?
1. $A B$ : defined because the number of columns of A is equal to the number of rows of B.
2. $A C$ : NOT defined, the number of columns of A is NOT equal to the number of rows of C.
3. $C E$ : defined because the number of columns of C is equal to the number of rows of E.
4. $E C$ : defined because the number of columns of E is equal to the number of rows of C.
5. $(A B) C$ : defined because AB is defined (see above) and the results is a matrix of order 2 by 5. The number of columns of AB is equal to 5 which is equal to the number of rows of C.
Part 2
1. $A B$ is defined and is given by
  $A B = [\begin{matrix} - 1 & 1 & - 2 \\ 0 & - 2 & 1 \end{matrix}] [\begin{matrix} - 1 & 2 & 0 \\ 0 & - 3 & 4 \\ - 1 & - 2 & 3 \end{matrix}] = [\begin{matrix} 3 & - 1 & - 2 \\ - 1 & 4 & - 5 \end{matrix}]$
2. $B C$ is not defined because the number of columns of B is not equal to the number of rows of C.
3. $A D$ is not defined because the number of columns of A is not equal to the number of rows of D.
4. $E F$ is not defined because the number of columns of E is not equal to the number of rows of F.
5. $F E$ is defined and is given by
  $F E = [\begin{matrix} - 1 & 0 & 2 \\ - 2 & - 3 & 4 \\ 1 & 4 & - 3 \end{matrix}] [\begin{matrix} 3 \\ 5 \\ - 11 \end{matrix}] = [\begin{matrix} - 25 \\ - 65 \\ 56 \end{matrix}]$
Part 3
Find the product $[\begin{matrix} x + y & - 2 \\ x - y & 1 \end{matrix}] [\begin{matrix} 2 & - 1 \\ 0 & - 2 \end{matrix}] = [\begin{matrix} 2 x + 2 y & - x - y + 4 \\ 2 x - 2 y & - x + y - 2 \end{matrix}]$
then solve
$[\begin{matrix} 2 x + 2 y & - x - y + 4 \\ 2 x - 2 y & - x + y - 2 \end{matrix}] = [\begin{matrix} 8 & 0 \\ 12 & - 8 \end{matrix}]$

Two matrices are equal if they have the same order and their corresponding entries are equal, hence the system of equations
$2 x + 2 y = 8, - x - y + 4 = 0, 2 x - 2 y = 12, - x + y - 2 = - 8$
Solve to obtain x = 5 and y = -1
Part 4
Calculate Rewrite the matrix as follows: $[\begin{matrix} 2 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{matrix}] = 2 [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$ Hence ${([\begin{matrix} 2 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{matrix}])}^{10} = 2^{10} {([\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}])}^{10}$ We note that the matrix $[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$ is an Row Operations and elementary matrix corresponding to interchanging rows 2 and 3. So to raise the elementary matrix to the power 10, we start with the elementary and interchange rows 2 and 3 9 times which gives the original matrix $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ Hence ${([\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}])}^{10} = 2^{10} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1024 & 0 & 0 \\ 0 & 1024 & 0 \\ 0 & 0 & 1024 \end{matrix}]$