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.
$$\left[\begin{array}{rrr}1 & 0 & 1 \\-3 & 1 & -2 \\0 & 0 & 1\end{array}\right] ;\left[\begin{array}{rrr}1 & 0 & -1 \\3 & 1 & -1 \\0 & 0 & 1\end{array}\right]$$

Key Concepts

Verification of Matrix Inverses

Verifying that two matrices are inverses involves calculating their product and checking whether the result is the identity matrix. This process confirms that one matrix undoes the transformation applied by the other. It is a fundamental method used in linear algebra to ensure the validity of inverse operations and the solutions of related equations.

Identity Matrix

The identity matrix is a special kind of square matrix characterized by ones along its main diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning any square matrix multiplied by the identity matrix (from either side) remains unchanged. The appearance of an identity matrix in the product of two matrices is a clear indicator that these matrices are inverses of each other.

Matrix Multiplication

Matrix multiplication is an operation where two matrices are combined to produce a third matrix. Each element in the resulting matrix is computed as the dot product of the corresponding row from the first matrix and the column from the second matrix. This operation is essential for analyzing relationships between linear transformations and is used in numerous applications, including verifying the properties of matrix inverses.

Inverse Matrix

An inverse matrix of a given square matrix is another matrix such that when the two matrices are multiplied together (in either order), the result is the identity matrix. Only nonsingular (invertible) matrices have inverses, and finding an inverse is a crucial step in solving linear systems and understanding how a linear transformation can be reversed.

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