Determine whether A and B are inverses by calculating AB and...

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator.
$$A=\left[\begin{array}{rrr}
-1 & -1 & -1 \\
4 & 5 & 0 \\
0 & 1 & -3
\end{array}\right] ; B=\left[\begin{array}{rrr}
15 & 4 & -5 \\
-12 & -3 & 4 \\
-4 & -1 & 1
\end{array}\right]$$

Key Concepts

Matrix Inverse

A matrix inverse is defined as a matrix that, when multiplied with the original matrix, produces the identity matrix. This concept is crucial because it provides a way to 'undo' the transformation represented by the matrix. Only square matrices that are nonsingular (i.e., have a nonzero determinant) possess an inverse, and finding this inverse is essential for solving systems of linear equations and understanding the properties of linear transformations.

Identity Matrix

The identity matrix is a special square matrix in which all the elements along the main diagonal are ones and all other elements are zeros. It serves as the multiplicative identity in matrix algebra, meaning that when any matrix is multiplied by the identity matrix, it remains unchanged. In the context of matrix inverses, if the product of two matrices results in the identity matrix, then each matrix is the inverse of the other.

Matrix Multiplication

Matrix multiplication is an operation in which the entries of the resulting matrix are computed as the dot products of the rows of the first matrix with the columns of the second matrix. This operation is not commutative, so the order of multiplication matters. When checking if two matrices are inverses, one must calculate both orders of multiplication (AB and BA) to ensure that each product is equal to the identity matrix, which validates the inverse relationship.

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