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Questions › IIT JAM › IIT JAM 2021 › Q36
2 marksMathematics › Matrices, determinants and linear systems › Solving linear systemsHardabout 3 min

IIT JAM 2021 Question 36: Matrices, determinants and linear systems

Let x,y∈Rx, y \in \mathbb{R}x,y∈R and M=[x+yx−yx−yx+y]M = \begin{bmatrix} x+y & x-y \\ x-y & x+y \end{bmatrix}M=[x+yx−y​x−yx+y​]. If M[31]=[−13]M\begin{bmatrix}3\\1\end{bmatrix} = \begin{bmatrix}-1\\3\end{bmatrix}M[31​]=[−13​], then (with adj(M)adj(M)adj(M) the adjoint and det⁡(M)\det(M)det(M) the determinant)

Select all that apply.

Answer and worked solution

Answer: (A) and (C)

The two rows give 4x+2y=−14x + 2y = -14x+2y=−1 and 4x−2y=34x - 2y = 34x−2y=3. Solving, x=14x = \tfrac14x=41​ and y=−1y = -1y=−1, so x+y=−34x + y = -\tfrac34x+y=−43​ (statement A) while x−y=54≠34x - y = \tfrac54 \neq \tfrac34x−y=45​=43​.

The determinant is det⁡(M)=(x+y)2−(x−y)2=4xy=4⋅14⋅(−1)=−1\det(M) = (x+y)^2 - (x-y)^2 = 4xy = 4\cdot\tfrac14\cdot(-1) = -1det(M)=(x+y)2−(x−y)2=4xy=4⋅41​⋅(−1)=−1, so (C) holds. For a 2×22\times22×2 matrix det⁡(adj(M))=det⁡(M)2−1=det⁡(M)=−1≠1\det(adj(M)) = \det(M)^{2-1} = \det(M) = -1 \neq 1det(adj(M))=det(M)2−1=det(M)=−1=1, so (D) is false. The correct statements are (A) and (C).

The trap: (D) det⁡(adj(M))=1\det(adj(M)) = 1det(adj(M))=1

For a 2×22\times22×2 matrix det⁡(adj(M))=det⁡(M)\det(adj(M)) = \det(M)det(adj(M))=det(M), which here is −1-1−1, not 111; assuming it is always positive gives the wrong statement (D).

Plain-text version

Let x, y ∈ ℝ and M = [x+y x-y; x-y x+y]. If M[3; 1] = [-1; 3], then (with adj(M) the adjoint and det (M) the determinant) (A) x + y = -3/4 (B) x - y = 3/4 (C) det (M) = -1 (D) det (adj(M)) = 1

Also written as: Let x, y in R and M = [x+y x-y; x-y x+y]. If M[3; 1] = [-1; 3], then (with adj(M) the adjoint and det (M) the determinant)

Matrices, determinants and linear systems has come up 102 times in 7 exams since 2006.

What this question tests

This question is on Solving linear systems, part of Matrices, determinants and linear systems in Mathematics.

Finding solutions of simultaneous linear equations by inversion, determinants or elimination.

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