Use partitioned matrices to prove by induction mat for n = 2,3,…, the n × n matrix A shown below is invertible and B is its inverse. A = [ 1 0 0 ⋯ 0 1 1 0 0 1 1 1 0 ⋮ ⋱ 1 1 1 ⋯ 1 ] B = [ 1 0 0 ⋯ 0 − 1 1 0 0 0 − 1 1 0 ⋮ ⋱ ⋱ 0 ⋯ − 1 1 ] For the induction step, assume A and B are ( k + 1) × ( k + 1) matrices, and partition A and B in a form similar to that displayed in Exercise 23.
Use partitioned matrices to prove by induction mat for n = 2,3,…, the n × n matrix A shown below is invertible and B is its inverse. A = [ 1 0 0 ⋯ 0 1 1 0 0 1 1 1 0 ⋮ ⋱ 1 1 1 ⋯ 1 ] B = [ 1 0 0 ⋯ 0 − 1 1 0 0 0 − 1 1 0 ⋮ ⋱ ⋱ 0 ⋯ − 1 1 ] For the induction step, assume A and B are ( k + 1) × ( k + 1) matrices, and partition A and B in a form similar to that displayed in Exercise 23.
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