Question

Let A=[(1,-1),(-2,3)] B=[(0,-2,1),(1,0,-4)] and C=[(3,0),(1,4),(-1,1)] be given matrices.Which of the following is true?

a. A⁻¹=A
b. AC is undefined
c. AB-2B is undefined
d. BC and CB have the same dimension

Answer 1

The statement AC is undefined is true because the dimensions of matrices A and C do not allow for multiplication. Other statements about A inverses, matrix subtraction, and dimension equivalence for BC and CB are false.

Step-by-step explanation:

The matrix operations and understanding the requirements for matrix multiplication, dimensions of matrices, and the concept of the inverse of a matrix. To determine whether the statements given are true or false, we consider each option:

• A⁻¹ = A: This is false, as the inverse of a matrix, when multiplied by the matrix itself, equals the identity matrix, not the original matrix.
• AC is undefined: This is true, because the number of columns in matrix A does not match the number of rows in matrix C, which is a requirement for matrix multiplication.
• AB - 2B is undefined: This is false, because while AB is not defined due to mismatched dimensions between A and B, 2B is a valid scalar multiplication, and thus subtraction with B is defined.
• BC and CB have the same dimension: This is false, because for matrix multiplication, the dimensions of the result depend on the outer dimensions of the multiplied matrices, and since B and C have different numbers of rows and columns, BC and CB will not even be defined and certainly won't have the same dimensions.