Question

if A=$\left[\begin{array}{ccc}i&0\\0&-i\end{array}\right]$ find A^2

Answer 1

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}i&0\\0&-i\end{array}\right]$

Now, Consider

$\rm :\longmapsto\: {A}^{2}$

$\rm \:  =  \: A \times A$

$\rm \:  =  \: \left[\begin{array}{ccc}i&0\\0&-i\end{array}\right] \times \left[\begin{array}{ccc}i&0\\0&-i\end{array}\right]$

$\rm \:  =  \: \left[\begin{array}{ccc} {i}^{2} + 0 &0 + 0\\0 + 0&0 + {( - i)}^{2} \end{array}\right]$

$\rm \:  =  \: \left[\begin{array}{ccc} {i}^{2} &0\\0& {i}^{2} \end{array}\right]$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ \: \: {i}^{2} \: = \: - 1 \: \: \: }}}$

So, using this, we get

$\rm \:  =  \: \left[\begin{array}{ccc} - 1&0\\0&-1\end{array}\right]$

Hence,

$\rm\implies \:\boxed{\tt{ \rm \:  {A}^{2} =  \: \left[\begin{array}{ccc} - 1&0\\0&-1\end{array}\right] \: \: }}$

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MORE TO KNOW

1. Matrix multiplication is defined when number of columns of pre multiplier is equal to the number of rows of post multiplier.

2. Matrix multiplication is Commutative, i e. AB = BA

3. Matrix multiplication is Associative, i.e. A(BC) = (AB)C

4. Matrix multiplication is Distributive, i.e. A(B + C) = AB + AC

5. There exist an identity matrix I such that AI = IA = I

Answer 2

We have been given a Matrix,

$\[\implies A=\left[\begin{array}{ccc}i&0\\0&-i\end{array}\right]\]$

With this information, we have been asked to calculate the value of A².

Evaluation :

$\implies {A}^{2} =A \times A$

$\implies {A}^{2} =\left[\begin{array}{ccc}i&0\\0&-i\end{array}\right] \times \left[\begin{array}{ccc}i&0\\0&-i\end{array}\right]$

$\implies {A}^{2} =\left[\begin{array}{ccc} {i}^{2} &0\\0& {i}^{2} \end{array}\right]$

We know that, ${i}^{2} = -1$. So, by substituting the value of ${i}^{2}$ in it, we get:

$\implies \boxed{{A}^{2} =\left[\begin{array}{ccc}- 1 &0\\0& - 1 \end{array}\right]}$

Hence, this is our required solution.

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LEARN MORE

Multiplication of Matrices.

Two matrices are conformable for multiplication if the number of columns matrix is equal to the number of rows of first matrix is equal to the number of rows of second matrix.

For example:

$\implies \rm{A = [a_{ij}]_{m \times n}}$

$\implies \rm{B = [b_{ij}]_{n \times p}}$

Then, AB = C. Where,

$\implies \rm{C = [b_{ij}]_{m \times p}}$