True or false and explain: 1) a). Non square matrices have inverses. b). Multipl

Question

True or false and explain: 1) a). Non square matrices have inverses. b). Multiplication of a matrix and its inverse is commutative. c). In a system of linear equations, if the determinate of the coefficient matrix is zero, the system has no solution. d). If you multiply two matrices and obtain the identity matrix, you can assume the matrices are inverses of each other. True or false and explain: 1) a). Non square matrices have inverses. b). Multiplication of a matrix and its inverse is commutative. c). In a system of linear equations, if the determinate of the coefficient matrix is zero, the system has no solution. d). If you multiply two matrices and obtain the identity matrix, you can assume the matrices are inverses of each other. 1) a). Non square matrices have inverses. b). Multiplication of a matrix and its inverse is commutative. c). In a system of linear equations, if the determinate of the coefficient matrix is zero, the system has no solution. d). If you multiply two matrices and obtain the identity matrix, you can assume the matrices are inverses of each other.

Explanation / Answer

a). Non-square matrices have inverses =======> FALSE

Because of the conditions on the inverse.

Suppose A^{-1} is the inverse of an n x m matrix A. Then we must have that
AA1=A1A=I

where I is the identity matrix.

But if A is n x m, then if we can multiply by A^{-1} on both the left and the right, A^{-1} must be m x n. But then
AA1=In and
A1A=Im
But these must be equal, and hence n=m and A is square.

b). Multiplication of a matrix and its inverse is commutative =====> TRUE

c). In a system of linear equations, if the determinate of the coefficient matrix is zero, the system has no solution =======> TRUE

If the determinant is nonzero than there exists exactly one solution. If the determinant is zero, there could be no solutions, or there could be infinitely many. It just means the matrix isn't invertible

d). If you multiply two matrices and obtain the identity matrix, you can assume the matrices are inverses of each other =======> TRUE

because it is precisely in the definition of the concept of inverse matrix
just read :
any (square) matrix A is said invertible
if and only if there exist B so that :
A * B = Id
and (both are needed because the product of matrices is not commutative)
B * A = Id
then B is unique and is noted : A^-1

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