(1) Without calculations, can you tell if the n x n matrix 1 2 ... 1 n – 1 ... 0 0 1 ... is invertible? Why?

B1. Advance maths Prove for an n by n invertible matrix we have A( Adj (A) ) = Det (A) (I), where I is n by n identity matrix.   Please show your work with using ‘induction’.

(3) Consider the matrix [1 2 3 -6 2 k 3] A : 4 For what values of k is A invertible?

Consider the following 4 x 4 matrix A: A = 0 1 3 0 3 5 -3 ܐ -2 -1 1 0 3 0 0 6 Now, let n ≥ 4 and m = n - 4. Then Im is the m x m identity matrix. Consider the nx n matrix B 0 (AL). B = Note that the 0's listed in the matrix B are actually matrices of all zeros. One will be a 4 x m matrix of all zeros and the other will be a m × 4 matrix of all zeros. Find the four fundamental subspaces of B. Explain everything; write out (nearly) every thought you have in a logical progression. Use complete sentences and proper grammar.