Problem 28 - Advanced Engineering Mathematics

Question

In Problems 27 and 28 , 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: Understanding the Problem

To find \((\mathbf{AB})^{-1}\), we need to determine the inverse of the product of matrices \(\mathbf{A}\) and \(\mathbf{B}\). Using the property of inverses in matrix multiplication, \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\). Both \(\mathbf{A}^{-1}\) and \(\mathbf{B}^{-1}\) are given, so we will compute the product in reverse order.

Step 2: Matrix Multiplication Setup

We have \(\mathbf{A}^{-1}\) and \(\mathbf{B}^{-1}\) given as:\[ \mathbf{A}^{-1} = \begin{pmatrix} 1 & 3 & -15 \ 0 & -1 & 5 \ -1 & -2 & 11 \end{pmatrix} \]\[ \mathbf{B}^{-1} = \begin{pmatrix} -1 & 1 & 0 \ 2 & 0 & 0 \ 1 & 1 & -2 \end{pmatrix} \]Next, we need to find the product \(\mathbf{B}^{-1}\mathbf{A}^{-1}\) by multiplying these two matrices.

Step 3: Perform Matrix Multiplication

To multiply \(\mathbf{B}^{-1}\) and \(\mathbf{A}^{-1}\), use the formula for the dot product of rows from the first matrix and columns from the second matrix:1. First row of \(\mathbf{B}^{-1}\) and columns of \(\mathbf{A}^{-1}\): - First element: \((-1)(1) + (1)(0) + (0)(-1) = -1\) - Second element: \((-1)(3) + (1)(-1) + (0)(-2) = -4\) - Third element: \((-1)(-15) + (1)(5) + (0)(11) = 20\)2. Second row of \(\mathbf{B}^{-1}\) and columns of \(\mathbf{A}^{-1}\): - First element: \((2)(1) + (0)(0) + (0)(-1) = 2\) - Second element: \((2)(3) + (0)(-1) + (0)(-2) = 6\) - Third element: \((2)(-15) + (0)(5) + (0)(11) = -30\)3. Third row of \(\mathbf{B}^{-1}\) and columns of \(\mathbf{A}^{-1}\): - First element: \((1)(1) + (1)(0) + (-2)(-1) = 3\) - Second element: \((1)(3) + (1)(-1) + (-2)(-2) = 2\) - Third element: \((1)(-15) + (1)(5) + (-2)(11) = -36\)

Step 4: Write the Resulting Product Matrix

After computing the dot products, the resulting matrix \(\mathbf{B}^{-1}\mathbf{A}^{-1}\) is:\[ \begin{pmatrix} -1 & -4 & 20 \ 2 & 6 & -30 \ 3 & 2 & -36 \end{pmatrix} \]This is \((\mathbf{AB})^{-1}\) according to the property used in Step 1.

Key Concepts

Matrix Multiplication

Matrix multiplication is a foundational concept in linear algebra. It's a way to combine two matrices to produce a third matrix, often revealing important properties about a system or transformation. To multiply two matrices, you need to align the rows of the first matrix with the columns of the second. The key steps are:

• **Dot Product:** Multiply each element of the row from the first matrix by the corresponding element of the column from the second matrix, then sum these products. This sum becomes an element of the resulting matrix.
• **Result Placement:** Place the resulting sum in the position corresponding to the initial row and column numbers being multiplied.

For example, given two matrices \mathbf{X} and \mathbf{Y}, you'd calculate the \(i,j\) element of the resulting matrix by taking the dot product of the \i^{th} row of \mathbf{X} with the \j^{th} column of \mathbf{Y}. It's crucial that the number of columns in the first matrix matches the number of rows in the second. If set up correctly, matrix multiplication can reveal intricate relationships between data sets or transform data in meaningful ways.

Inverse Properties

The inverse of a matrix, denoted as \(\mathbf{A}^{-1}\), is a matrix that, when multiplied with the original matrix \(\mathbf{A}\), yields the identity matrix. This concept is akin to finding the reciprocal of a number in arithmetic. Here are some important properties of matrix inverses:

• **Existence:** Not every matrix has an inverse. A matrix must be square (same number of rows and columns) and its determinant must be non-zero for an inverse to exist.
• **Inverse of a Product:** The inverse of the product of two matrices, say \mathbf{AB}, is the product of the inverses of \mathbf{A} and \mathbf{B} taken in reverse order: \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\).
• **Uniqueness:** The inverse of a matrix, if it exists, is unique.

Understanding these properties helps when solving matrix equations or finding solutions to systems of linear equations efficiently. When dealing with matrices like \mathbf{A}^{-1} and \mathbf{B}^{-1}, these properties make solving for \(\mathbf{(AB)}^{-1}\) straightforward by simply respecting the order of multiplication.

Linear Algebra

Linear algebra is a branch of mathematics concentrating on vector spaces and linear mappings between them. It's a highly applicable field, providing tools and frameworks for complex problem-solving. In the context of this exercise involving matrix inverses, linear algebra principles can simplify and streamline finding solutions to intricate problems.

• **Vector Spaces:** Central to linear algebra, these spaces consist of vectors, which can be scaled and added together to form new vectors.
• **Transformations:** Matrices can serve as linear transformations, rotating, scaling, or otherwise changing vectors while respecting the linearity constraints.
• **Systems of Equations:** One of the key applications of matrices and their inverses is solving systems of linear equations efficiently. Armed with matrix arithmetic and inverse properties, one can find solutions that might be cumbersome or impossible with traditional algebraic methods.

Linear algebra provides a language for much of the advanced work in mathematics, physics, engineering, and computer science, illustrating its core significance and far-reaching impact.