<p>The number of right inverses for the matrix \(\begin{bmatrix} 1 & -1 & 2 \\ 2 & -1 & 1 \end{bmatrix}\) is</p>
Step-by-Step Solution
Key Concept: A matrix A has right inverses if and only if it has full row rank (rank equals number of rows). The number of right inverses is infinite when they exist, determined by the dimension of the null space of A.
<p><strong>Step 1:</strong> Check if right inverse exists. For a 2×3 matrix A to have a right inverse, it must have full row rank (rank = 2).</p><p><strong>Step 2:</strong> Find rank of A = [1 -1 2; 2 -1 1]. Apply row operations: R₂ → R₂ - 2R₁ gives [1 -1 2; 0 1 -3]. Rank = 2 ✓</p><p><strong>Step 3:</strong> Since rank(A) = 2 = number of rows, right inverses exist. If B is a right inverse, then AB = I₂.</p><p><strong>Step 4:</strong> A 2×3 matrix A with rank 2 has infinitely many right inverses because the equation AX = I₂ represents 4 equations in 6 unknowns (entries of X). The solution space has dimension 6 - 4 = 2, giving a 2-parameter family of solutions.</p><p><strong>Step 5:</strong> Alternatively, if rank = m (rows) and n (columns) with m < n, the number of right inverses is infinite (uncountably many).</p><p>∴ Answer: D (Infinite/Infinitely many)</p>
Correct Answer: D