Problem 5 - College Algebra with Modeling and Visualization

Question

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{rrr} 2 & 1 & -1 \\ 3 & 0 & 2 \\ -1 & 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 0 & 1 & -2 \\ 1 & -3 & 7 \\ 0 & -1 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

Yes, \(B\) is the inverse of \(A\) because \(AB = I\) and \(BA = I\).

Step 1: Define the Operations

To determine if matrix \(B\) is the inverse of matrix \(A\), we need to calculate the product \(AB\) and \(BA\). If both products result in the identity matrix, then \(B\) is the inverse of \(A\). The identity matrix for a 3x3 matrix is \(I = \left[\begin{array}{rrr}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\).

Step 2: Compute Matrix Product \(AB\)

Calculate the product \(AB\) by multiplying matrix \(A\) by matrix \(B\):\(AB = \begin{bmatrix}2 & 1 & -1 \ 3 & 0 & 2 \ -1 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}0 & 1 & -2 \ 1 & -3 & 7 \ 0 & -1 & 3\end{bmatrix} = \begin{bmatrix}(2\cdot0 + 1\cdot1 + -1\cdot0) & (2\cdot1 + 1\cdot-3 + -1\cdot-1) & (2\cdot-2 + 1\cdot7 + -1\cdot3) \ (3\cdot0 + 0\cdot1 + 2\cdot0) & (3\cdot1 + 0\cdot-3 + 2\cdot-1) & (3\cdot-2 + 0\cdot7 + 2\cdot3) \ (-1\cdot0 + 0\cdot1 + 1\cdot0) & (-1\cdot1 + 0\cdot-3 + 1\cdot-1) & (-1\cdot-2 + 0\cdot7 + 1\cdot3)\end{bmatrix} \)Simplifying each entry, we get:\(AB = \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix} = I\)

Step 3: Compute Matrix Product \(BA\)

Calculate the product \(BA\) by multiplying matrix \(B\) by matrix \(A\):\(BA = \begin{bmatrix}0 & 1 & -2 \ 1 & -3 & 7 \ 0 & -1 & 3\end{bmatrix} \cdot \begin{bmatrix}2 & 1 & -1 \ 3 & 0 & 2 \ -1 & 0 & 1\end{bmatrix} \)Continue with the multiplication:\(= \begin{bmatrix}(0\cdot2 + 1\cdot3 + -2\cdot-1) & (0\cdot1 + 1\cdot0 + -2\cdot0) & (0\cdot-1 + 1\cdot2 + -2\cdot1) \ (1\cdot2 + -3\cdot3 + 7\cdot-1) & (1\cdot1 + -3\cdot0 + 7\cdot0) & (1\cdot-1 + -3\cdot2 + 7\cdot1) \ (0\cdot2 + -1\cdot3 + 3\cdot-1) & (0\cdot1 + -1\cdot0 + 3\cdot0) & (0\cdot-1 + -1\cdot2 + 3\cdot1)\end{bmatrix} \)Simplifying each entry, we have:\(BA = \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix} = I\)

Step 4: Conclusion: Confirm Inverse

Since both \(AB\) and \(BA\) equal the identity matrix \(I\), matrix \(B\) is indeed the inverse of matrix \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. This ensures that each element in the resulting matrix is derived from the sum of products:

• The entry in the first row and first column is calculated by multiplying each element of the first row in the first matrix by the corresponding element in the first column of the second matrix and adding them up.
• This process is repeated for each element in the resulting matrix.

Matrix multiplication is not commutative, meaning that AB does not necessarily equal BA. This is crucial when checking for inverse matrices. It is, however, associative and distributive. Understanding these properties can help solve complex matrix-related problems with ease. Remember always to align dimensions, since incorrectly set matrices cannot be multiplied.

Identity Matrix

The identity matrix plays a critical role in matrix algebra, much like the number 1 does in basic arithmetic. For a matrix to have an inverse, there must exist an identity matrix such that when multiplied with its inverse, results in the identity matrix itself:

• An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere.
• For a 3x3 identity matrix, it's represented as \[\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

For any square matrix A, if there exists another matrix B such that both AB and BA are the identity matrix, B is considered the inverse of A. The identity matrix is vital in confirming this relationship, and understanding its form and function is key when working with inverse matrices.

Matrix Algebra

Matrix algebra amalgamates rules and operations that guide how matrices are used together. It encompasses operations like matrix addition, scalar multiplication, and the more complex operations of multiplication and finding inverses.
Key rules of matrix algebra include:

• Associative Rule: (AB)C = A(BC)
• Distributive Rule: A(B + C) = AB + AC
• Identity Element: For any matrix A, A multiplied by the identity matrix I results in A.

Understanding how these operations work together helps when solving equations involving matrices. It’s crucial to follow these algebraic rules to correctly perform procedures like solving systems of linear equations or analyzing matrix transformations.