If A is a square matrix such that A Adj - A - beginbmatrix 4 - 0 - 0 10 - 4 - 0 10 - 0 - 411lendbmatrix - then -adj adj- A - adj-A is equal toA- 256B- 16C- 64D- 32

The correct option is B 16 A(adj. A) = |A|. I_n Clearly, |A| = 4, n = 3 |adj. (adj. A)| = |A|^(n 1)^2 = 4^4 = 256 |adj. A| = |A|^n 1 = 4^2 = 16 ∴|adj.(adj A)|/ ...

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Byju's Answer

Standard XII

Mathematics

Addition and Subtraction of a Matrix

If A is a squ...

Question

If A is a square matrix such that $A (A d j . A) = ⎡ ⎢ ⎣ \begin{matrix} 4 & 0 & 0 0 & 4 & 0 0 & 0 & 4 \end{matrix} ⎤ ⎥ ⎦,$ then $\frac{| a d j (a d j . A) |}{| a d j . A |}$ is equal to

256

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Solution

The correct option is D
16

$A (a d j . A) = | A | . I_{n}$

Clearly, $| A | = 4, n = 3$

$| a d j . (a d j . A) | = | A |^{(n - 1)^{2}} = 4^{4} = 256$

$| a d j . A | = | A |^{n - 1} = 4^{2} = 16$

$∴ \frac{| a d j . (a d j A) |}{| (a d j . A) |} = \frac{256}{16} = 16$

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Q. Let

A

be a square matrix such that

A (a d j A) = ⎡ ⎢ ⎣ \begin{matrix} 4 & 0 & 0 0 & 4 & 0 0 & 0 & 4 \end{matrix} ⎤ ⎥ ⎦ .

Then the value of

\frac{| a d j (a d j A) |}{| a d j A |}

Q. If A,B,C are three square matrices of third order such that

A = ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 2 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦

| B | = 2^{2} 3^{2}, | C | = 2, x, y, z \in I^{+}

| a d j (a d j A B C) | = 2^{16} 3^{8} 7^{4}

then number of distinct possible matrices A is

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦

, then

A^{n} =

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If A is a square matrix such that A(Adj.A)=⎡⎢⎣400040004⎤⎥⎦, then |adj(adj.A)||adj.A| is equal to

The correct option is D 16 A(adj.A)=|A|.In Clearly, |A|=4,n=3 |adj.(adj.A)|=|A|(n−1)2=44=256 |adj.A|=|A|n−1=42=16 ∴|adj.(adjA)||(adj.A)|=25616=16

If A is a square matrix such that $A (A d j . A) = ⎡ ⎢ ⎣ \begin{matrix} 4 & 0 & 0 0 & 4 & 0 0 & 0 & 4 \end{matrix} ⎤ ⎥ ⎦,$ then $\frac{| a d j (a d j . A) |}{| a d j . A |}$ is equal to

The correct option is D
16

$A (a d j . A) = | A | . I_{n}$

Clearly, $| A | = 4, n = 3$

$| a d j . (a d j . A) | = | A |^{(n - 1)^{2}} = 4^{4} = 256$

$| a d j . A | = | A |^{n - 1} = 4^{2} = 16$

$∴ \frac{| a d j . (a d j A) |}{| (a d j . A) |} = \frac{256}{16} = 16$