Question

If A and B are square matrices of order 2 such that |-4AB|=-256 & |A|=-2, then |B| is equal to: * I will mark u as brainlist if u will give me right answer,I am confused in this question​

Answer 1

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

A is a square matrix of order 2

B is a square matrix of order 2

| - 4 A B | = - 256

| A | = - 2

$\large\underline{\sf{To\:Find - }}$

$\boxed{ \bf{ \: |B| }}$

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

Given that,

A is a square matrix of order 2

B is a square matrix of order 2

| - 4 AB | = - 256

| A | = - 2

Now, Consider

$\rm :\longmapsto\: | - 4AB| = \: - \: 256$

We know that,

If A is a square matrix of order n, then

$\boxed{ \bf{ \: |kA| = {k}^{n} |A|}}$

So, using this identity, we get

$\rm :\longmapsto\: {( - 4)}^{2} |AB| = - \: 256$

$\rm :\longmapsto\: 16 |AB| = - \: 256$

$\rm :\longmapsto\: |AB| = \: - \: 16$

We know, that,

If A and B are square matrices of order n, then

$\boxed{ \bf{ \: |AB| = |A| |B| }}$

So, using this identity, we get

$\rm :\longmapsto\: |A| |B| = - \: 16$

$\rm :\longmapsto\: - 2 |B| = \: - \: 16$

$\bf\implies \: |B| \: \: = \: \: 8$

Additional Information :-

$\boxed{ \bf{ \: |adj \: A| = { |A| }^{n - 1}}}$

$\boxed{ \bf{ \: |A \: adj \: A| = { |A| }^{n}}}$

$\boxed{ \bf{ \: | {A}^{ - 1} | = \frac{1}{ |A| }}}$

$\boxed{ \bf{ \: |I| = 1}}$

$\boxed{ \bf{ \: A \: (adj \: A) = (adj \: A)A = |A|I}}$

$\boxed{ \bf{ \: {AA}^{ - 1} = {A}^{ - 1}A = I}}$