Let A be a 2 × 2 matrix with det(A) = -1 and det <math> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>+</mo> <mi>I</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mi>d</mi> <mi>j</mi> <mrow> <mo>(</mo> <mrow> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>I</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>.</mo> </mrow> </math> Then the sum of the diagonal elements of A can be:

| (A + I) (adj A + I)| = 4 Þ |A adj A + A + Adj A + I| = 4 Þ | (A)I + A + adj A + I|= 4|A| = -1 Þ |A + adj A| = 4 A = [ a b c d ] ⇒ a d j A = [ a − b − c d ] ⇒ | ( a + d ) 0 0 ( a + d ) | = 4 ⇒ a + d = ± 2

Let A be a 2 × 2 matrix with det(A) = -1 and det <math> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>+</mo> <mi>I</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mi>d</mi> <mi>j</mi> <mrow> <mo>(</mo> <mrow> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>I</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>.</mo> </mrow> </math> &nbsp;Then the sum of the diagonal elements of A can be:

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Maths Class 12th Maths Determinants

Let A be a 2 × 2 matrix with det(A) = -1 and det $((A + I) (A d j (A) + I)) = 4 .$ Then the sum of the diagonal elements of A can be:

Option 1 -

-1

Option 2 -

Option 3 -

Option 4 -

$- \sqrt[]{2}$

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1 Answer

Answered by

Raj Pandey | Contributor-Level 9

2 weeks ago

Correct Option - 2

Detailed Solution:

| (A + I) (adj A + I)| = 4 Þ |A adj A + A + Adj A + I| = 4 Þ | (A)I + A + adj A + I|= 4|A| = -1

Þ |A + adj A| = 4

$A = [\begin{matrix} a & b \\ c & d \end{matrix}] \Rightarrow a d j A = [\begin{matrix} a & - b \\ - c & d \end{matrix}] \Rightarrow | \begin{matrix} (a + d) & 0 \\ 0 & (a + d) \end{matrix} | = 4 \Rightarrow a + d = \pm 2$

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Let A be a 2 × 2 matrix with det(A) = -1 and det $((A + I) (A d j (A) + I)) = 4 .$ Then the sum of the diagonal elements of A can be:

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Let A be a 2 × 2 matrix with det(A) = -1 and det ( ( A + I ) ( A d j ( A ) + I ) ) = 4 . Then the sum of the diagonal elements of A can be:

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Let A be a 2 × 2 matrix with det(A) = -1 and det $((A + I) (A d j (A) + I)) = 4 .$ Then the sum of the diagonal elements of A can be: