Problem 2 - Advanced Engineering Mathematics

Question

In Problems 1 and 2 , verify that the matrix \(\mathbf{B}\) is the inverse of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 0 \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrr} 2 & -1 & 2 \\ 1 & -1 & 2 \\ -3 & 2 & -3 \end{array}\right) $$

Step-by-Step Solution

Answer

Matrix \( \mathbf{B} \) is the inverse of matrix \( \mathbf{A} \), since \( \mathbf{AB} = \mathbf{BA} = \mathbf{I} \).

Step 1: Understand the Criteria for Inverse Matrices

To verify that matrix \( \mathbf{B} \) is the inverse of matrix \( \mathbf{A} \), we need to show that the product of \( \mathbf{A} \) and \( \mathbf{B} \) is the identity matrix \( \mathbf{I} \).

Step 2: Compute the Product AB

Calculate \( \mathbf{AB} = \mathbf{A} \times \mathbf{B} \). Using the given matrices:\[\mathbf{A} = \begin{pmatrix} 1 & -1 & 0 \ 3 & 0 & 2 \ 1 & 1 & 1 \end{pmatrix},\mathbf{B} = \begin{pmatrix} 2 & -1 & 2 \ 1 & -1 & 2 \ -3 & 2 & -3 \end{pmatrix}\]Multiply the first row of \( \mathbf{A} \) by each column of \( \mathbf{B} \) to get the first row of \( \mathbf{AB} \). Continue this process for all rows. The result is:\[\mathbf{AB} = \left(\begin{array}{rrr}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{array}\right)\]

Step 3: Interpret the Result

The resulting matrix \( \mathbf{AB} \) is the identity matrix \( \mathbf{I} \), which confirms that \( \mathbf{B} \) is indeed the inverse of \( \mathbf{A} \).

Step 4: Verify Product BA

Although finding \( \mathbf{AB} = \mathbf{I} \) suffices, let's verify by computing \( \mathbf{BA} \) as well, to confirm it's also the identity matrix:\[\mathbf{BA} = \mathbf{B} \times \mathbf{A} = \begin{pmatrix} 2 & -1 & 2 \ 1 & -1 & 2 \ -3 & 2 & -3 \end{pmatrix} \times \begin{pmatrix} 1 & -1 & 0 \ 3 & 0 & 2 \ 1 & 1 & 1 \end{pmatrix}\]Multiplying similarly as before, we obtain:\[\mathbf{BA} = \left(\begin{array}{rrr}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{array}\right)\]

Step 5: Conclude from Both Products

Since both \( \mathbf{AB} = \mathbf{I} \) and \( \mathbf{BA} = \mathbf{I} \), matrix \( \mathbf{B} \) is indeed the inverse of matrix \( \mathbf{A} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices to produce another matrix. The operation is not as simple as multiplying individual elements, but rather involves a process where elements of the rows of the first matrix are multiplied with elements of the columns of the second matrix and then summed. This process is repeated for all rows and columns, resulting in a new matrix. It is important to note that matrix multiplication is not commutative, meaning that \(\mathbf{A} \times \mathbf{B} eq \mathbf{B} \times \mathbf{A}\) in general.

In the context of verifying inverses, we multiply the matrix \(\mathbf{A}\) by its supposed inverse \(\mathbf{B}\), expecting to get the identity matrix \(\mathbf{I}\). For example, for a \(3 \times 3\) matrix, as in our exercise, each element of the row of \(\mathbf{A}\) is multiplied by every element of the column of \(\mathbf{B}\), and the products are summed to form one element of the resulting matrix. This computation helps confirm whether two matrices are inverses if their multiplication equals the identity matrix.

Identity Matrix

The identity matrix is a special kind of square matrix that plays a similar role to the number 1 in scalar multiplication. For an \(n \times n\) identity matrix, denoted as \(\mathbf{I}_n\), the diagonal elements are all 1s, and all other elements are 0s. This matrix serves as the neutral element in matrix multiplication, meaning that multiplying any matrix \(\mathbf{A}\) by the identity matrix \(\mathbf{I}\), results in \(\mathbf{A}\): \(\mathbf{A} \times \mathbf{I} = \mathbf{A}\).

In verifying that \(\mathbf{B}\) is the inverse of \(\mathbf{A}\), the product \(\mathbf{A} \times \mathbf{B}\) is expected to result in the identity matrix. If this condition is met, it means that applying \(\mathbf{A}\) followed by \(\mathbf{B}\) (or vice versa) to any vector will leave that vector unchanged, thus confirming that the operations of these matrices cancel each other out, similar to how multiplying a number by its reciprocal results in 1.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. It forms the foundation for many areas in mathematics and applied sciences. One of its core concepts involves matrices and their operations, such as addition, subtraction, and multiplication.

A significant application in linear algebra is solving systems of linear equations, which can often be represented as matrix equations. Understanding inverses is essential, as they are used to solve matrix equations of the form \(\mathbf{A}\mathbf{x} = \mathbf{b}\). If \(\mathbf{A}\) is invertible, the solution for \(\mathbf{x}\) is given by \(\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}\). This highlights the importance of verifying the inverse of a matrix, as done in the exercise, to ensure that systems of equations can be solved efficiently.