Visual SolutionPYQ 2024 · Apr 4 Shift 2Standard

Question

Let

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

and

B = I + \text{adj}(A) + (\text{adj}\,A)^2 + \ldots + (\text{adj}\,A)^{10}

. Then the sum of all elements of the matrix

B

is:

(A)

-124

(B)

22

(C)

-88

(D)

-110

Solution Path

\text{adj}(A) = \begin{bmatrix}1&-2\\0&1\end{bmatrix}

. Power pattern:

(\text{adj}\,A)^k = \begin{bmatrix}1&-2k\\0&1\end{bmatrix}

. Sum

B = \begin{bmatrix}11&-110\\0&11\end{bmatrix}

. Element sum

= -88

01Question Setup

1/4

Find the sum of all elements of

B = I + \text{adj}(A) + (\text{adj}\,A)^2 + \cdots + (\text{adj}\,A)^{10}

where

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

02Find Adjoint

2/4

For a

2 \times 2

matrix, swap the diagonal entries and negate the off-diagonal entries.

\text{adj}(A) = \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix}

03Power PatternKEY INSIGHT

3/4

By induction (or direct multiplication),

(\text{adj}\,A)^k = \begin{bmatrix} 1 & -2k \\ 0 & 1 \end{bmatrix}

. The off-diagonal scales linearly.

(\text{adj}\,A)^k = \begin{bmatrix} 1 & -2k \\ 0 & 1 \end{bmatrix}

04Final Answer

4/4

B = \begin{bmatrix} 11 & -2(0+1+\cdots+10) \\ 0 & 11 \end{bmatrix} = \begin{bmatrix} 11 & -110 \\ 0 & 11 \end{bmatrix}

. Sum

= 11 - 110 + 0 + 11 = -88

\boxed{-88}

Concepts from this question2 concepts unlocked

★

STANDARD

Every square matrix satisfies its own characteristic equation: if p(x) = |A - xI|, then p(A) = 0.

A^2 - \text{tr}(A) \cdot A + |A| \cdot I = 0 \quad (\text{for } 2 \times 2)

Reduces higher powers of A to lower ones. Essential for expressing A⁻¹ in terms of A and I.

Matrix inverseHigher powers of AMatrix polynomial

EASY

det(AB) = det(A) × det(B). Also det(Aᵀ) = det(A) and det(kA) = kⁿ det(A) for n×n matrix.

|AB| = |A| \cdot |B|, \quad |A^T| = |A|, \quad |kA| = k^n|A|

JEE loves chained matrix products like det(ABA^T). This rule avoids explicit multiplication entirely.

Product of matricesTranspose determinantScalar multiplication

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