Let A - begin bmatrix -2 - 1113 - 4 e n d b m a t r i x -Then verify A adj A -adj A A - A - I- where I is the identity matrix of order 2 -

We've A=[ 2 amp; 1; 3 amp; 4 ]⇒adj.A A=[ 4 amp; 3; 1 amp; 2 ]^T=We've A=[ 4 amp; 1; 3 amp; 2 ] ∴A(adj A) =[ 2 amp; 1; ...

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Byju's Answer

Standard XII

Mathematics

Identity Matrix

Let A = begin...

Question

Let $A = [\begin{matrix} - 2 & 1 3 & 4 \end{matrix}]$ .
$Then verify A(adj A) = (adj A)A = |A|I, where I is the identity matrix of order 2$ .

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Solution

$We've A = [\begin{matrix} - 2 & 1 3 & 4 \end{matrix}] \Rightarrow adj.A A = {[\begin{matrix} 4 & - 3 - 1 & - 2 \end{matrix}]}^{T} = We've A = [\begin{matrix} 4 & - 1 - 3 & - 2 \end{matrix}] ∴ A(adj A) = [\begin{matrix} - 2 & 1 3 & 4 \end{matrix}] [\begin{matrix} 4 & - 1 - 3 & - 2 \end{matrix}] = [\begin{matrix} - 11 & 0 0 & - 11 \end{matrix}] = - 11 [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] . . . . . (i) ∴ (adj A)A = [\begin{matrix} 4 & - 1 - 3 & - 2 \end{matrix}] [\begin{matrix} - 2 & 1 3 & 4 \end{matrix}] = [\begin{matrix} - 11 & 0 0 & - 11 \end{matrix}] = - 11 [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] . . . . . (i i) Also, | A | = ∣ ∣ ∣ [\begin{matrix} - 2 & 1 3 & 4 \end{matrix}] ∣ ∣ ∣ = - 8 - 3 = - 11..... (i i i) By (i),(ii) & (iii), A(adj A) = (adj A)A = |A|I is verified .$

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Similar questions

Let $A = [\begin{matrix} 2 & 4 3 & 2 \end{matrix}], B = [\begin{matrix} 1 & 3 - 2 & 5 \end{matrix}], C = [\begin{matrix} - 2 & 5 3 & 4 \end{matrix}]$ . Find each of the following:
(i)A+B

(ii)A-B

(iii)3A-C

(iv)AB

(v)BA

Show that $(i) [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}] [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] \neq [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}]$

$(i i) ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ \neq ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$

Q. Let

A = [\begin{matrix} 2 & 3 1 & 2 \end{matrix}] and B = A = [\begin{matrix} 4 & - 6 - 2 & 4 \end{matrix}] . Then compute AB. Hence, solve the following system of equations : 2x + y = 4, 3x + 2y = 1 .

Q. Let A denote the matrix

[\begin{matrix} 0 & i i & 0 \end{matrix}]

, where

i^{2} = - 1

and lit I denote the identity matrix

[\begin{matrix} 0 & i i & 0 \end{matrix}]

.Then

I + A + A^{2} + . . . . . A^{2010}

Q. Verify

A (a d j A) = (a d j A) A = | A | I

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 3 & 0 & 2 1 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦

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Let A=[−2134]. Then verify A(adj A) = (adj A)A = |A|I, where I is the identity matrix of order 2.

Let $A = [\begin{matrix} - 2 & 1 3 & 4 \end{matrix}]$ .
$Then verify A(adj A) = (adj A)A = |A|I, where I is the identity matrix of order 2$ .