Problem 28 - Advanced Engineering Mathematics

Question

Use the given matrices to find \((\mathbf{A B})^{-1}\). $$ \mathbf{A}^{-1}=\left(\begin{array}{rrr} 1 & 3 & -15 \\ 0 & -1 & 5 \\ -1 & -2 & 11 \end{array}\right), \quad \mathbf{B}^{-1}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 2 & 0 & 0 \\ 1 & 1 & -2 \end{array}\right) $$

Step-by-Step Solution

Answer

\((\mathbf{AB})^{-1} = \begin{pmatrix} -1 & -4 & 20 \\ 2 & 6 & -30 \\ 3 & 2 & -36 \end{pmatrix}\)

Step 1: Understand the Problem

We need to find the inverse of the product of two matrices, \((\mathbf{AB})^{-1}\). We are given the inverses of the individual matrices \(\mathbf{A}\) and \(\mathbf{B}\).

Step 2: Apply the Properties of Inverses

Recall the property \((\mathbf{AB})^{-1} = \mathbf{B}^{-1} \mathbf{A}^{-1}\). This property tells us that to find the inverse of a product of two matrices, we can multiply the inverses in the reverse order.

Step 3: Multiply the Inverse Matrices

Compute \(\mathbf{B}^{-1} \mathbf{A}^{-1}\) using matrix multiplication:\[\mathbf{B}^{-1} = \begin{pmatrix} -1 & 1 & 0 \ 2 & 0 & 0 \ 1 & 1 & -2 \end{pmatrix}\]\[\mathbf{A}^{-1} = \begin{pmatrix} 1 & 3 & -15 \ 0 & -1 & 5 \ -1 & -2 & 11 \end{pmatrix}\]

Step 4: Perform Matrix Multiplication

Multiply the matrices \(\mathbf{B}^{-1}\) and \(\mathbf{A}^{-1}\):1. Calculate the first row: - First column: \((-1)(1) + (1)(0) + (0)(-1) = -1\) - Second column: \((-1)(3) + (1)(-1) + (0)(-2) = -4\) - Third column: \((-1)(-15) + (1)(5) + (0)(11) = 20\)2. Calculate the second row: - First column: \((2)(1) + (0)(0) + (0)(-1) = 2\) - Second column: \((2)(3) + (0)(-1) + (0)(-2) = 6\) - Third column: \((2)(-15) + (0)(5) + (0)(11) = -30\)3. Calculate the third row: - First column: \((1)(1) + (1)(0) + (-2)(-1) = 3\) - Second column: \((1)(3) + (1)(-1) + (-2)(-2) = 2\) - Third column: \((1)(-15) + (1)(5) + (-2)(11) = -36\)Thus, \(\mathbf{B}^{-1} \mathbf{A}^{-1} = \begin{pmatrix} -1 & -4 & 20 \ 2 & 6 & -30 \ 3 & 2 & -36 \end{pmatrix}\).

Step 5: Conclusion

The inverse of the product of the matrices \(\mathbf{A}\) and \(\mathbf{B}\), \((\mathbf{AB})^{-1}\), is given by the matrix \(\begin{pmatrix} -1 & -4 & 20 \ 2 & 6 & -30 \ 3 & 2 & -36 \end{pmatrix}\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra. It involves taking two matrices and producing a third matrix by taking the sum of the products of entries across corresponding rows and columns. For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. Here’s how it works:

• When multiplying matrix \(\mathbf{B}^{-1}\) by matrix \(\mathbf{A}^{-1}\), take each row of \(\mathbf{B}^{-1}\) and pair it with each column of \(\mathbf{A}^{-1}\).
• Multiply the corresponding entries and take the sum—this gives one entry in the product matrix.
• Repeat this process for each combination of rows and columns to fill in the entire product matrix.

Matrix multiplication is not commutative, meaning that \(\mathbf{A} \cdot \mathbf{B} eq \mathbf{B} \cdot \mathbf{A}\) in general, so the order matters.

Properties of Inverse Matrices

The properties of inverse matrices are essential in simplifying and solving matrix equations. A matrix \(\mathbf{A}\) has an inverse \(\mathbf{A}^{-1}\) if it meets specific conditions, like being a square matrix and having a non-zero determinant. Key properties of inverse matrices include:

• Inverse of a Product: The inverse of a product of matrices \((\mathbf{AB})^{-1}\) is equal to the inverse of \(\mathbf{B}\) multiplied by the inverse of \(\mathbf{A}\) in reverse order: \((\mathbf{AB})^{-1} = \mathbf{B}^{-1} \mathbf{A}^{-1}\).
• Associative Property: Matrix inverses follow associative laws, meaning you can regroup them during multiplication: \(\mathbf{A}(\mathbf{B} \mathbf{C}) = (\mathbf{A} \mathbf{B})\mathbf{C}\).
• Identity Matrix: Multiplying any matrix by its inverse results in the identity matrix, \( \mathbf{A} \mathbf{A}^{-1} = \mathbf{I} \).

Understanding these properties helps to simplify the operation of finding the inverse of a product of matrices, as shown in the given exercise.

Linear Algebra Concepts

Linear algebra concepts underpin many of the calculations and operations in mathematics related to matrices. Some of the fundamental ideas include:

• Determinants: A matrix must have a non-zero determinant to have an inverse. This serves as a check before attempting to find an inverse.
• Identity Matrix: The matrix equivalent of the number 1 in multiplication, where any matrix \(\mathbf{A}\) multiplied by the identity matrix \(\mathbf{I}\) remains unchanged: \(\mathbf{A} \mathbf{I} = \mathbf{A}\).
• System of Linear Equations: Matrices are often used to solve systems of linear equations by representing them in matrix form and finding solutions through techniques like Gaussian elimination or using inverses.
• Rank and Dimension: The rank of a matrix determines the dimension of the vector space spanned by its rows or columns, which affects the matrix's properties, including invertibility.

By mastering these linear algebra concepts, you can better understand how matrices behave and interact, giving you powerful tools for solving mathematical problems.