Falak Kinney

2021-01-28

Multiply the given matrix. After performing the multiplication, describe what happens to the elements in the first matrix. $\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

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Step 1
Given: $\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Step 2
Let matrix $A=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]$ & matrix $B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
we have to find the multiplication of matrices A&B that means we have to find $A×B$
We know the formula for multiplication of $2×2$ matrices.
$\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]\left[\begin{array}{cc}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}{b}_{11}+{a}_{12}{b}_{21}& {a}_{11}{b}_{12}+{a}_{12}{b}_{22}\\ {a}_{21}{b}_{11}+{a}_{22}{b}_{21}& {a}_{22}{b}_{12}+{a}_{22}{b}_{22}\end{array}\right]$
Step 3
Here from matrix A we get,
${a}_{11}={a}_{11},{a}_{12}={a}_{12},{a}_{21}={a}_{21},{a}_{22}={a}_{22}$
from matrix B we get,
${b}_{11}=1,{b}_{12}=0,{b}_{21}=0,{b}_{22}=1$
Thus matrix multiplication $A×B$ becomes,
$\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\left({a}_{11}×1\right)+\left({a}_{12}×0\right)& \left({a}_{11}×0\right)+\left({a}_{12}×1\right)\\ \left({a}_{21}×1\right)+\left({a}_{22}×0\right)& \left({a}_{22}×0\right)+\left({a}_{22}×1\right)\end{array}\right]$
$=\left[\begin{array}{cc}{a}_{11}+0& 0+{a}_{12}\\ {a}_{21}+0& 0+{a}_{22}\end{array}\right]$
$=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]$
Step 4
Therefore we get,
$A×B=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]$
Thus after multiplication of matrices A & B we get again the matrix which equals A.
After multiplication of two matrices the elements in the first matrix remains same.

Jeffrey Jordon