Name: Matrix Multiplication and Associated Properties
Uploaded: 2019-07-19
Duration: 6 min 22 s
Description: We learned the addition and subtraction of matrices. Is it possible to multiply the two matrices? Watch this video and learn to multiply two matrices together.

Question

If

A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]

,

B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]

and

AB = BA

, then show that

B

is a scalar matrix.

Accepted Answer

Given that,

$A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ , $B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$

To proof : $B$ is a scalar matrix

Now,

$AB = BA$

$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$

$[\begin{matrix} 1 \times a + 2 \times 0 & 1 \times 0 + 2 \times b \\ 3 \times a + 4 \times 0 & 3 \times 0 + 4 \times b \end{matrix}] = [\begin{matrix} 1 \times a + 2 \times 0 & 1 \times 0 + 2 \times b \\ 3 \times 0 + 4 \times 0 & 3 \times 0 + 4 \times b \end{matrix}]$

$[\begin{matrix} a + 0 & 0 + 2 b \\ 3 a + 0 & 0 + 4 b \end{matrix}] = [\begin{matrix} a + 0 & 2 a + 0 \\ 0 + 3 b & 0 + 4 b \end{matrix}]$

$[\begin{matrix} a & 2 b \\ 3 a & 4 b \end{matrix}] = [\begin{matrix} a & 2 a \\ 3 b & 4 b \end{matrix}]$

Thus their corresponding entries are equal.

$∴ 2 b = 2 a \Rightarrow b = a$

And

$3 a = 3 b \Rightarrow a = b$

Hence, $B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ is a scalar matrix since all the diagonal elements are equal and all non-diagonal entries are zero.

If $A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ , $B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ and $AB = BA$ , then show that $B$ is a scalar matrix.

Important Questions on Matrices

Learn and Prepare for any exam you want !

If A=1234, B=a00b and AB=BA, then show that B is a scalar matrix.

Important Questions on Matrices

Learn and Prepare for any exam you want !

If $A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ , $B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ and $AB = BA$ , then show that $B$ is a scalar matrix.