Q.For the matrix A and its inverse found in Q1, verify that AA−1=A−1A=I.
Concept understanding — Invertible Matrices and Uniqueness of Inverse
A square matrix A of order n is invertible (non-singular) if some square matrix B of the same order satisfies AB=BA=I; such a B is called the inverse of A, denoted A−1. A complete proof, resting only on the associativity of matrix multiplication, shows that this inverse -- when it exists -- is always unique: if B and C both satisfy the defining condition, then B=BI=B(AC)=(BA)C=IC=C, so B=C. This uniqueness is what justifies the single notation A−1 without ambiguity. For a 2×2 matrix A=\begin{bmatrix}a&b\c&d\end{bmatrix}, the practical shortcut A−1=ad−bc1[d−b\-ca] applies whenever ad−bceq0; if ad−bc=0, A is singular and no inverse exists (and any matrix that is a zero-divisor is automatically singular, since an inverse would force the other zero-divisor factor to be the zero matrix, a contradiction).
[!TLDR] Multiply A by the computed A−1 in both orders. [!ANSWER] AA−1=A−1A=[1001]=I.
A=[3121], A−1=[1−1−23]. For AA−1: position (1,1): 3(1)+2(−1)=1; (1,2): 3(−2)+2(3)=−6+6=0; (2,1): 1(1)+1(−1)=0; (2,2): 1(−2)+1(3)=1. So AA−1=[1001]=I. For A−1A: (1,1): 1(3)+(−2)(1)=1; (1,2): 1(2)+(−2)(1)=0; (2,1): −1(3)+3(1)=0; (2,2): −1(2)+3(1)=1. So A−1A=[1001]=I as well. [!ANSWER] AA−1=A−1A=I.
Multiply A by the computed A−1 in both possible orders using the row-by-column rule, and confirm both products equal the identity matrix exactly.
Checking only one of AA−1 or A−1A and assuming the other automatically holds is a shortcut that should be avoided in a first careful verification, even though for genuine inverses both do hold.
- CBSE 2025Set ANNUAL1 markMCQQ.Matrices A and B will be inverse of each other only if(a) AB = BA(b) AB = BA = O(c) AB = O, BA = I(d) AB = BA = I
›Reveal solutionSolution
Two square matrices are inverses of each other exactly when both products give the identity matrix.
By definition, if A and B are square matrices of the same order such that AB=BA=I (the identity matrix), then B is called the inverse of A (written B=A−1) and A is the inverse of B. A single-sided product, or a product equal to the zero matrix, does not establish an inverse relationship.
✓Final answer(d) AB=BA=I.
- CBSE 2025Set ANNUAL1 markMCQQ.If the matrix (4 -k; -2 3) has no inverse matrix, then the value of k is(a) 6(b) -6(c) 12(d) -12
›Reveal solutionSolution
A matrix has no inverse exactly when its determinant is zero.
A square matrix is invertible only if it is non-singular, i.e. det=0. "No inverse matrix" means the matrix IS singular, so we need det=0.
det(4−2−k3)=4(3)−(−k)(−2)=12−2k
Set this to zero: 12−2k=0⇒k=6.
✓Final answerk=6 (option a).
- CBSE 2024Set ANNUAL1 markMCQQ.Matrices A and B are invertible of each other iff(a) AB=BA(b) AB=BA=0(c) AB=0,BA=I(d) AB=BA=I
›Reveal solutionSolution
By definition, A and B are inverses of each other exactly when their product (in either order) gives the identity matrix.
A square matrix B is the inverse of A if AB=BA=I, where I is the identity matrix of the same order. This is the standard definition of matrix invertibility - both products must equal the identity, not zero or any other matrix.
✓Final answer(d) AB=BA=I.
- CBSE 2023Set ANNUAL1 markMCQQ.Matrices A and B will be inverse of each other only if-(a) AB=BA(b) AB=BA=0(c) AB=0,BA=I(d) AB=BA=I
›Reveal solutionSolution
Inverse of a matrix is defined by the identity relation on both sides.
By definition, if A is a square matrix and there exists a matrix B of the same order such that AB=BA=I (the identity matrix), then B is called the inverse of A (written B=A−1), and A is the inverse of B. Both products must equal the identity matrix, not zero or in only one order.
✓Final answerOption (d) AB=BA=I
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