Examine the product of the two matrices to determine if each...

Examine the product of the two matrices to determine if each is the inverse of the other.
$$\left[\begin{array}{rrr}1 & 0 & -1 \\3 & 1 & -1 \\0 & 0 & 0\end{array}\right] ;\left[\begin{array}{rrr}1 & 0 & -1 \\-3 & 1 & -2 \\0 & 0 & 1\end{array}\right]$$

Key Concepts

Matrix Multiplication

Matrix multiplication is the process of combining two matrices to form a new matrix by summing the products of corresponding entries. It is essential in verifying matrix inverses since the definition of an inverse relies on the product of a matrix and its inverse yielding the identity matrix. Correctly performing matrix multiplication, considering the dimensions and proper alignment of rows and columns, is critical when checking for invertibility.

Identity Matrix

The identity matrix is a special type of square matrix that has ones on its main diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. In verifying whether two matrices are inverses of each other, the resulting product must match the identity matrix, thereby confirming that each matrix undoes the effect of the other.

Matrix Inverse

The matrix inverse is a fundamental concept in linear algebra, where a square matrix A has an inverse A?¹ if their product in both orders gives the identity matrix. This operation is analogous to division for numbers, and it requires that the matrix be non-singular (i.e., have a non-zero determinant). In the context of a product check, one verifies that the candidate matrix multiplied by the given matrix results in the identity matrix, confirming that each is the inverse of the other.

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