Question

If A is a non-singular square matrix of order 3 such that |A| = 3, then value of |2AT| is

Answer 1

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

A is a non singular matrix of order 3, such that |A| = 3

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

$\rm \: |2 {A}^{T} |$

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

Given that,

A is a non - singular matrix of order 3 such that |A| = 3

Now, Consider

$\rm \: |2 {A}^{T} |$

We know, In A is a square matrix of order n and k is non zero real number, then

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

So, using this result, we get

$\rm \: = \: {2}^{3} | {A}^{T} |$

We know, if A is a square matrix of order n, then

$\boxed{\tt{ \: \: | {A}^{T} | = |A| \: \: }}$

So, using this, we get

$\rm \: = \: 8 |A|$

$\rm \: = \: 8 \times 3$

$\rm \: = \: 24$

Hence,

$\rm\implies \: |2 {A}^{T} | = 24$

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ADDITIONAL INFORMATION

If A is a square matrix of order n, then

$\rm \: |adjA| \: = \: { |A| }^{n - 1}$

$\rm \: |A \: adjA| \: = \: { |A| }^{n}$

$\rm \: adj(adjA) = { |A| }^{n - 2} \: A$

$\rm \: adj(adjA) = { |A| }^{(n - 1)^{2} }$

$\rm \: A \: is \: singular \: matrix, \: \rm\implies \: |A| = 0$

$\rm \: A \: is \: non \: - \: singular \: matrix, \: \rm\implies \: |A| \ne 0$

$\rm \: {A}^{ - 1} \: exist \: \rm\implies \: |A| \ne 0$

$\rm \: | {A}^{ - 1} | = \dfrac{1}{ |A| }$

$\rm \: |AB| = |A| \: |B|$

$\rm \: | {A}^{n} | = { |A| }^{n}$