Question

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}-1 & 0 & 2 \\ 1 & -2 & 0 \\ 1 & 0 & 3\end{array}\right], B=\frac{1}{10}\left[\begin{array}{rrr}-6 & 0 & 4 \\ -3 & -5 & 2 \\ 2 & 0 & 2\end{array}\right]\)

Short answer

Yes, B is the inverse of A as their product AB results in an identity matrix.

Step-by-step solution

Understand the Concept of Inverse Matrix

In matrix algebra, when the product of two matrices (say, A and B) yields the identity matrix, B is said to be the inverse of A.

Perform Matrix Multiplication - Multiply Matrix A by B

Use the rules of matrix multiplication to multiply Matrix A by Matrix B. Each element of the result matrix is obtained by multiplying each row of the first matrix by each column of the second matrix and summing up the products.

Check if Result is Identity Matrix

The resulting matrix from Step 2 should be the identity matrix. The identity matrix of order 3 is \[I_3 = \left[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\]. If the matrix multiplication from Step 2 resulted in this matrix, then B is indeed the inverse of A.

Key concepts

Matrix Algebra

In the realm of mathematics, matrix algebra is a pivotal subject, especially when we venture into linear algebra. A matrix is essentially an array of numbers arranged in rows and columns. Consider it much like a spreadsheet, where each data entry fits into its own cell. But unlike spreadsheets, matrices follow very specific rules for operations such as addition, subtraction, and multiplication.

For students, grappling with matrix operations can be somewhat daunting at first. However, with practice and a clear understanding of the rules, it becomes second nature. When working with matrices, make sure to ensure that the operations are valid. For instance, you can only add or subtract two matrices if they have the same dimensions, and when multiplying two matrices, the number of columns in the first must match the number of rows in the second.

Matrix algebra is also the bedrock for more advanced topics such as determinants, rank, and, as our exercise shows, finding the inverse of a matrix. The inverse of a matrix is a foundational concept, which plays a crucial role in solving systems of equations, among other applications.

Matrix Multiplication

Matrix multiplication might seem like a direct extension of basic arithmetic, but it has its unique set of rules. To multiply two matrices together, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The product matrix's dimension will have the same number of rows as the first and the same number of columns as the second.

The process involves taking the rows of the first matrix and the columns of the second matrix to calculate the entries of the product matrix. Specifically, you take the dot product of each row and column pair, which means multiplying the corresponding elements and then adding up those products. This can be succinctly described by the formula for the element in the ith row and jth column of the product matrix: \(C_{ij} = A_{i1}B_{1j} + A_{i2}B_{2j} + ... + A_{in}B_{nj}\).

For our exercise, we multiply Matrix A by Matrix B, using the properties of matrix multiplication to determine the resulting matrix. If done correctly, the resulting product should honor the intricate dance of matrix multiplication, demonstrating the coherent structure that makes matrix algebra so powerful.

Identity Matrix

The identity matrix is the neutral element in the world of matrix multiplication. Just like multiplying any number by one yields the same number, multiplying any matrix by the identity matrix of appropriate dimensions will yield the same matrix.

The identity matrix is a square matrix, which means it has the same number of rows as columns. Its main diagonal (which runs from the top-left corner to the bottom-right corner) is composed entirely of ones, and all other entries are zeros. Symbolically, it is often denoted by \(I_n\) where n represents the dimensions of the square matrix. For example, the identity matrix of size 3, denoted as \(I_3\), is:
\[I_3 = \left[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\]

In our context, confirming that Matrix B is the inverse of Matrix A means verifying that when A is multiplied by B, the result is indeed the identity matrix. This check is a definitive litmus test for Matrix B's candidature as the inverse of Matrix A.