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  3. problem 3 it is often useful to write a matrix...

Question: problem 3 it is often useful to write a matrix...

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Problem 3. It is often useful to write a matrix as an array of smaller matrices. For example, 1,112),A11 E Rixni,A12 RmiXn2, A2,1 R2xni, A22 Rm2Xm2 2,1 2,2 a. We define the following block matrices 1,1 2,1 1,2 2 2,2 Show that rank(A) S rank(C1) rank(C2). Find examples for which the above inequality is strict. b. Consider two matrices 1,2 2,2 BiA B22 1,1 1,2 2,1 2,1 Find the condition for AB -C to be defined and When matrix A is a square matrix, it is also more useful to select the diagonal blocks to be square matrices. Show that in general detA # detA1,1 detA2,2-det(A1,2A2,1) and det A det A 1,1 detA 2.2 ) c. - det(A2.1A1.2 d. Show that when A1.1 is invertible, we have det A det A1,1 det(A22 - A2,1A11A1,2

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