solve all marked question, linear algebra In Exercises 9 and 10, mark each state

Question

solve all marked question, linear algebra

In Exercises 9 and 10, mark each statement True or False. Justify each answer. a. In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true. B. If A and B are n times n and invertible, then A^-1 B^-1 is the inverse of AB. c. If A = [a b c d] and ab - cd notequalto 0, then A is invertible. If A is an invertible n times n matrix, then the equation AX = b is consistent for each b in R^n. e. Each elementary matrix is invertible. a. A product of invertible n times n matrices is invertible, and the inverse of the product is the product of their inverses in the same order. b. If A is invertible, then the inverse of A^-1 is A itself. c. If A = [a b c d] and ad = bc, then A is not invertible. d. If A can be row reduced to the identity matrix, then A must be invertible. e. If A is invertible, then elementary row operations that reduce A to the identity I_n also reduce A^-1 to I_n. Let A be an invertible n times n matrix, and let B be an n times p matrix. Show that the equation AX = B has a unique solution A^-1 B.

Explanation / Answer

9

d.

True

If A is invertible then the columns of A are linearly independent and hence span Rn

So any b is in span of columns of Rn

Hence, for each b in Rn we have an x so that

Ax=b

11.

B has p nx1 size columns

Each columns is in span of col(A)

So let xi be the coefficient vector for bi ie ith columns of B

So that

Axi=bi

Hence, AX=B

where, X has the ith column xi

Hence proved

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