Definition 1. A square matrix A is called a diagonal matrix if all the off-diagonal entries...

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Definition 1. A square matrix A is called a diagonal matrix if all the off-diagonal entries of A are zero. That is, aij = 0, wherever i j. If the diagonal entries of A are a₁, a2,..., an, we write A = diag(a1, a2..., an). (a) Let A be the diagonal matrix A = diag(2, -4,1). Use Gauss-Jordan Elimination to find A-¹, (b) Suppose that A is an n x n diagonal matrix with A = diag(a1, a2..., an) and suppose that all the diagonal entries are non-zero. Generalize your work in part (a) to find a formula for A-1, and prove that your formula is in fact the inverse matrix of A. Definition 1. A square matrix A is called a diagonal matrix if all the off-diagonal entries of A are zero. That is, aij = 0, wherever i j. If the diagonal entries of A are a₁, a2,..., an, we write A = diag(a1, a2..., an). (a) Let A be the diagonal matrix A = diag(2, -4,1). Use Gauss-Jordan Elimination to find A-¹, (b) Suppose that A is an n x n diagonal matrix with A = diag(a1, a2..., an) and suppose that all the diagonal entries are non-zero. Generalize your work in part (a) to find a formula for A-1, and prove that your formula is in fact the inverse matrix of A.