Problem 3 - Precalculus: Mathematics for Calculus

Question

Verifying the Inverse of a Matrix Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\) \(A=\left[\begin{array}{ll}4 & 1 \\ 7 & 2\end{array}\right] B=\left[\begin{array}{rr}2 & -1 \\ -7 & 4\end{array}\right]\)

Step-by-Step Solution

Answer

Matrix \(B\) is the inverse of matrix \(A\) because both \(AB\) and \(BA\) equal the identity matrix.

Step 1: Calculate the Product AB

To find the product of matrices \(A\) and \(B\), we multiply each element of the row of matrix \(A\) by the corresponding element of the column of matrix \(B\) and then sum these products.\[AB = \begin{pmatrix} 4 & 1 \ 7 & 2 \end{pmatrix} \times \begin{pmatrix} 2 & -1 \ -7 & 4 \end{pmatrix}\]Let's calculate each element of the resulting 2x2 matrix:- First row, first column: \((4 \times 2) + (1 \times -7) = 8 - 7 = 1\)- First row, second column: \((4 \times -1) + (1 \times 4) = -4 + 4 = 0\)- Second row, first column: \((7 \times 2) + (2 \times -7) = 14 - 14 = 0\)- Second row, second column: \((7 \times -1) + (2 \times 4) = -7 + 8 = 1\)Thus, the product \(AB\) is:\[AB = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]

Step 2: Calculate the Product BA

Similarly, to verify \(B\) is the inverse of \(A\), we need to calculate \(BA\) using the same method:\[BA = \begin{pmatrix} 2 & -1 \ -7 & 4 \end{pmatrix} \times \begin{pmatrix} 4 & 1 \ 7 & 2 \end{pmatrix}\]Calculate each element:- First row, first column: \((2 \times 4) + (-1 \times 7) = 8 - 7 = 1\)- First row, second column: \((2 \times 1) + (-1 \times 2) = 2 - 2 = 0\)- Second row, first column: \((-7 \times 4) + (4 \times 7) = -28 + 28 = 0\)- Second row, second column: \((-7 \times 1) + (4 \times 2) = -7 + 8 = 1\)Thus, \(BA\) is:\[BA = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]

Step 3: Confirmation of Inverse

We have calculated both products \(AB\) and \(BA\), and both are equal to the identity matrix \(I\):\[I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]Since both products equal the identity matrix, matrix \(B\) is indeed the inverse of matrix \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation essential for various mathematical computations. It involves taking two matrices and producing a new matrix. Each element in the resulting matrix is the sum of products of elements from the rows of the first matrix and columns of the second matrix. Let's break it down step by step for clarity:

• Consider two matrices, say matrix A with dimensions m x n and matrix B with dimensions n x p. The resulting product matrix will have dimensions m x p.
• To calculate an element in the resulting matrix, say at position (i, j), we multiply each element of the i-th row of matrix A by the corresponding element of the j-th column of matrix B. Then, we sum all these products.

The key is to ensure that the number of columns in the first matrix matches the number of rows in the second matrix for multiplication to be possible. This is crucial in confirming if two matrices can indeed be multiplied.

Identity Matrix

An identity matrix plays a vital role in matrix operations similar to the number 1 in multiplication. It's called an identity matrix because, when multiplied by any compatible matrix, it leaves the other matrix unchanged, thereby maintaining the identity of that matrix. Here's what makes an identity matrix unique:

• An identity matrix is always square, meaning it has the same number of rows and columns.
• On the main diagonal (from the top left to the bottom right), each element is 1, while all other elements are 0.

  Example:\[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]

In matrix multiplication, if you multiply any matrix by an identity matrix of appropriate size, you get the original matrix back. This property of maintaining the original matrix's identity is essential in verifying inverse matrices.

Inverse Matrix Verification

To verify that one matrix is the inverse of another, it's essential to perform specific calculations. These calculations help confirm that matrices truly serve as each other's inverses.

• The main task is to compute both the product AB and the product BA, where A and B are the matrices in question.
• If both products result in an identity matrix, it confirms that one matrix is the inverse of the other.
• In essence, if \(A B = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\) and \(B A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\), then matrix B is the inverse of matrix A and vice versa.

This approach hinges on the special nature of the identity matrix and is crucial in many applications such as solving linear equations, computer graphics, and more. Understanding this verification process not only solidifies one's grasp on matrix theory but also deepens comprehension of how systems of equations can be solved efficiently.