True or False? In Exercises 45 and 46, determine whether each statement is true

Question

True or False? In Exercises 45 and 46, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 45. (a) The inverse of a nonsingular matrix is unique. (b) If the matrices A, B, and C satisfy BA-CA and A is invertible, then B = C (c) The inverse of the product of two matrices is the product of their inverses; that is, (AB)A-B- (d) If A can be row reduced to the identity matrix, then A is nonsingular. 46, (a) The product of four invertible 7 × 7 matrices is invertible (b) The transpose of the inverse of a nonsingular matrix is equa (c) The matrix is invertible if ab-dc0. (d) If A is a square matrix, then the system of linear equations Ax = b has a unique solution.

Explanation / Answer

45 (a) The inverse of a nonsingular matrix is unique. TRUE

REASON- If A has inverses B and C, then B=C.

If A has an inverse, it is denoted by A -1 = In = A-1 A.

Also if A is non-singular ,then A-1 is also non-singular then

(A-1)-1 =A.

45 (b ). This true

reason- .If A is invertible you multiply both sides of the equation by A inverse to get IB=IC which simplifies to get B =C.

45(c). False

Reason- If A,B be the two matrices ,then the reciprocal of their product is (AB)-1.

clearly, (AB). (B-1.A-1) = A (BB)-1A-1

   = AIA-1=AA-1=I

similarly, (B-1.A-1). (AB) = I

HENCE , B-1.A-1 IS THE RECIPROCAL OF AB.

i.e, (AB)-1 = B-1. A-1.

45(d). True

46(a) False.

It is invertible, but the inverses in the product of inverses in the rverse order.

46(b) false

The transpose of the product of the two matrices is their transposes taken in reverse order.

46(c) False

A is invertible is ad not equal to 0.

46(d) True

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