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 & 0 \\ 0 & 1 \end{matrix}]

and

B = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]

, show that

(aA + bB) (aA - bB) = (a^{2} + b^{2}) A

.

Accepted Answer

Given that,

$A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and $B = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$

To proof : $(aA + bB) (aA - bB) = (a^{2} + b^{2}) A$

Now,

$(aA + bB) (aA - bB) = (a [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + b [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]) (a [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] - b [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}])$

$(aA + bB) (aA - bB) = ([\begin{matrix} a & 0 \\ 0 & a \end{matrix}] + [\begin{matrix} 0 & b \\ - b & 0 \end{matrix}]) ([\begin{matrix} a & 0 \\ 0 & a \end{matrix}] - [\begin{matrix} 0 & b \\ - b & 0 \end{matrix}])$

$(aA + bB) (aA - bB) = [\begin{matrix} a & b \\ - b & a \end{matrix}] [\begin{matrix} a & - b \\ b & a \end{matrix}]$

$(aA + bB) (aA - bB) = [\begin{matrix} a \times a + b \times b & a \times (- b) + b \times a \\ (- b) \times a + a \times b & (- b) \times (- b) + a \times a \end{matrix}]$

$(aA + bB) (aA - bB) = [\begin{matrix} a^{2} + b^{2} & - ab + ab \\ - ab + ab & b^{2} + a^{2} \end{matrix}]$

$(aA + bB) (aA - bB) = [\begin{matrix} a^{2} + b^{2} & 0 \\ 0 & b^{2} + a^{2} \end{matrix}]$

$(aA + bB) (aA - bB) = a^{2} + b^{2} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Hence, $(aA + bB) (aA - bB) = (a^{2} + b^{2}) A$ .

If $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and $B = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$ , show that $(aA + bB) (aA - bB) = (a^{2} + b^{2}) A$ .

Important Questions on Matrices

Learn and Prepare for any exam you want !

If A=1001 and B=01-10, show that aA+bBaA-bB=a2+b2A.

Important Questions on Matrices

Learn and Prepare for any exam you want !

If $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and $B = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$ , show that $(aA + bB) (aA - bB) = (a^{2} + b^{2}) A$ .