Suppose
𝐴
is an
𝑛
×
𝑛
invertible matrix. Which of the following statements is FALSE?

Question

Suppose 
𝐴
 is an 
𝑛
×
𝑛
 invertible matrix. Which of the following statements is FALSE?

BlackTom AI · Accepted Answer

We begin by restating the problem to ensure clarity: Given an invertible n×n matrix A, identify which statement is FALSE among the following options.

Option 1: 'Ax = b has a unique solution for all vectors b in R^n.' This is true because A is invertible, so for any b there exists a unique x = A^{-1} b solving Ax = b. The key idea is that invertibility guarantees both existence and uniqueness of the solution for every right-hand side vector.

Option 2: 'For any n×n matrix B, we have A^2 B A^{-1} = A B.' To test this, rewrite the equation by multiplying on the left by A^{-1}: A B A^{-1} = B. This would require that B commutes with A for all B, which is only true in the special case where A is a scalar multiple of the identity. In general, a fixed invertible A does not commute with every B, so the statement is false in general. The flaw is assuming universal commutativity between A and all matrices B, which contradicts standard linear algebra results about non-scalar invertible matrices.

Option 3: 'A^T A is an invertible matrix.' Since A is invertible, det(A^T A) = det(A^T) det(A) = det(A)^2 ≠ 0, so A^T A is invertible. The underlying concept is that the product of a matrix with its transpose has determinant equal to the square of det(A), ensuring invertibility when A is invertible.

Option 4: 'A^{-1} = (A^T A)^{-1} A^T.' This is a standard matrix identity: (A^T A)^{-1} A^T equals A^{-1} when A is invertible, provided we interpret the product as (A^T A)^{-1} times A^T. A quick check is to multiply both sides by A on the right: (A^T A)^{-1} A^T A = (A^T A)^{-1} (A^T A) = I, which is consistent with A^{-1} A = I. The algebra shows the equality holds, so the statement is true.

Adding a cross-check for understanding, notice that the crux of the issue in Option 2 is relying on a universal commutativity property that does not hold in general. Options 1, 3, and 4 either follow directly from invertibility or from standard identities involving transposes and inverses, without requiring extra assumptions about B or A beyond invertibility.

Suppose 𝐴 is an 𝑛 × 𝑛 invertible matrix. Which of the following statements is FALSE?Single choice

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