Question

if A is non-singular square matrix of oder 3 such that A^2 = 3A , then value of |A| is . what will be the answer ?

please give me the answer ❓​

Answer 1

Given :  A is non-singular square matrix of order 3 such that A² = 3A

To Find : value of |A|

Solution:

A² = 3A

=> A.A = 3A

A is non-singular square matrix  Hence A⁻¹ Exist

Multiply both sides by A⁻¹

=> A.AA⁻¹ = 3AA⁻¹

AA⁻¹ = I

=> A . I  = 3I

A . I   = A

=> A = 3I

square matrix of order 3  

Hence

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

A = 3I

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

 $A= \left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right]$

| A|  = 3 (3 * 3 - 0) - 0  + 0    =  27

Hence value of |A| is 27

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Answer 2

SOLUTION

GIVEN

A is non-singular square matrix of oder 3 such that A² = 3A

TO DETERMINE

The value of |A|

FORMULA TO BE IMPLEMENTED

If A is a non-singular square matrix of oder n then

$\sf1. \: \: |kA| = {k}^{n} |A|$

$\sf 2. \: |AB| = | A || B |$

EVALUATION

Here it is given that ,

A is non-singular square matrix of oder 3 such that A² = 3A

Now we have

$\sf {A}^{2} = 3A$

Taking determinant in both sides we get

$\sf | { A}^{2} | = | 3A |$

$\sf \implies \: |A.A| = | 3A |$

$\sf \implies \: |A.A| = {3}^{3} |A | \: \: ( \: Using \: Formula \: 1 \: )$

$\sf \implies \: {|A|}^{2} = {3}^{3} |A | \: \: ( \: Using \: Formula \: 2 \: )$

$\sf \implies \: {|A|}^{2} = 27 |A |$

$\sf \implies \: {|A|}^{} = 27 \: \: \: \: \: \: (\because \: |A | \ne 0 \: )$

FINAL ANSWER

The value of |A| = 27

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