Let A be an n x n invertible matrix such that A = LU, where L...

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Let A be an n x n invertible matrix such that A = LU, where L is a unit lower triangular matrix and U is an upper triangular matrix. For 1 ≤ k ≤n, let A denote the principal leading submatrix of A formed by the first k rows and the first k columns of A. Show that det (Ak) 0, k = 1,2,..., n. (3 marks) Let A be an n x n invertible matrix such that A = LU, where L is a unit lower triangular matrix and U is an upper triangular matrix. For 1 ≤ k ≤n, let A denote the principal leading submatrix of A formed by the first k rows and the first k columns of A. Show that det (Ak) 0, k = 1,2,..., n. (3 marks)

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