Question

QQ.if a=(3 0/5 1) and b=(-4 2/1 0) find A^2-2ab+b^2​

Answer 1

Step-by-step explanation:

Given:

$\begin{gathered}A=\left[\begin{array}{cc}3&0\\5&1\end{array}\right]\\\end{gathered}$

and

$\begin{gathered}B=\left[\begin{array}{cc}-4&2\\1&0\end{array}\right]\\\end{gathered}$

To Find:A²-2AB+B²

Solution:

We know that

$\begin{gathered}\bold{A ^{2} -2AB+B^{2} = ( {A -B) }^{2}} \\ \end{gathered}$

Step 1: Find A-B

$\begin{gathered}A-B=\left[\begin{array}{cc}3&0\\5&1\end{array}\right]-\left[\begin{array}{cc}-4&2\\1&0\end{array}\right]\\\end{gathered}$

$\begin{gathered}A-B=\left[\begin{array}{cc}3+4&0-2\\5-1&1-0\end{array}\right]\\\end{gathered}$

$\begin{gathered}A-B=\left[\begin{array}{cc}7&-2\\4&1\end{array}\right]\\\end{gathered}$

Step 2: Find (A-B)²

$\begin{gathered}{(A-B)}^2=\left[\begin{array}{cc}7&-2\\4&1\end{array}\right]\times \left[\begin{array}{cc}7&-2\\4&1\end{array}\right] \\\end{gathered}$

$</p><p>\begin{gathered}{(A-B)}^2=\left[\begin{array}{cc}7(7)+(-2)(4)&7(-2)+(-2)(1)\\4(7)+1(4)&4(-2)+1(1)\end{array}\right]\\\end{gathered}$

$\begin{gathered}{(A-B)}^2=\left[\begin{array}{cc}41&-16\\32&-7\end{array}\right]\\\end{gathered}$

Final answer:

$\begin{gathered}\bold{\red{A ^{2} -2AB+B^{2} =\left[\begin{array}{cc}41&-16\\32&-7\end{array}\right]}}\\\end{gathered}$

Hope it helps you.

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