Problem 15 - Algebra and Trigonometry

Question

use the fact that if \(\boldsymbol{A}=\left[\begin{array}{ll}{\boldsymbol{a}} & {\boldsymbol{b}} \\ {\boldsymbol{c}} & {\boldsymbol{d}}\end{array}\right],\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc}{d} & {-b} \\ {-c} & {a}\end{array}\right]\) to find the inverse of each matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$ A=\left[\begin{array}{rr} {3} & {-1} \\ {-4} & {2} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix A is \[\begin{array}{cc}{1} & {0.5} \ {2} & {1.5}\end{array}\]

Step 1: Find the determinant

The determinant of matrix A (denoted as ad - bc) can be calculated using the elements of A. We have a = 3, b = -1, c = -4 and d = 2. Therefore, the determinant is (3*2) - (-1*-4) = 6 - 4 = 2. Since the determinant is not equal to zero, we can find an inverse of the matrix.

Step 2: Find the inverse using the given formula

Substitute a, b, c and d with respective values in the inverse matrix formula. The inverse of A is given by \(\frac{1}{2}\times\)\[\begin{array}{cc}{2} & {1} \ {4} & {3}\end{array}\] which simplifies to \[\begin{array}{cc}{1} & {0.5} \ {2} & {1.5}\end{array}\]

Step 3: Verify the inverse

Now we need to verify that AA⁻¹ and A⁻¹A both equate to the identity matrix. When we multiply A with its calculated inverse, we get \[\begin{array}{cc}{1} & {0} \ {0} & {1}\end{array}\], which is the identity matrix. The same holds true when we multiply A⁻¹ with A. Thus, we have verified the correctness of the calculated inverse.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can tell us various properties of the matrix. It plays a crucial role when it comes to understanding if a matrix has an inverse. Let's explore how to calculate it for a 2x2 matrix, which is a simple yet important type of matrix.

Given a 2x2 matrix \( \boldsymbol{A} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant (denoted as \( \det(A) \) or just \( ad - bc \)) is found using the formula:

• Multiply the top left and bottom right entries: \( a \times d \).
• Multiply the top right and bottom left entries: \( b \times c \).
• Subtract the second product from the first: \( ad - bc \).

In our example, matrix \( A \) has entries \( a = 3 \), \( b = -1 \), \( c = -4 \), and \( d = 2 \). The determinant is calculated as \( 3 \times 2 - (-1) \times (-4) = 6 - 4 = 2 \).

If the determinant is zero, the matrix does not have an inverse. But here, since the determinant is 2, we can proceed to find the inverse of matrix \( A \).

Identity Matrix

The identity matrix is a very special matrix because when it is multiplied with any matrix that it aligns with, it leaves the other matrix unchanged. It is much like the number 1 for multiplication in arithmetic. It's the 'do-nothing' but essential building block in linear algebra.

The identity matrix \( I_{2} \) for 2x2 matrices is written as:

• \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

In the exercise, we used the identity matrix to verify the correctness of our calculated inverse matrix. To do this, we multiply our original matrix \( A \) by its calculated inverse \( A^{-1} \) and checked that the result is indeed the identity matrix. This confirms that our inverse calculations were done correctly.

Also, multiplying the inverse \( A^{-1} \) by \( A \) again gave us the identity matrix, showing that \( A^{-1} \) truly is the inverse of \( A \).

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra and involves combining matrices in a way that yields another matrix. Although it might initially seem complicated, it follows a consistent pattern:

When multiplying matrices, like in our scenario for verifying inverses, follow these steps:

• Consider matrices \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and \( B = \begin{bmatrix} e & f \ g & h \end{bmatrix} \).
• To find each element of the resulting matrix, take the dot product of rows from \( A \) and columns from \( B \).
• For example, the top left element of the result is \( ae + bg \).
• Continue this method for each position in the resultant matrix.

In our exercise, we did matrix multiplication to verify the inverse of matrix \( A \).

By multiplying \( A \) and \( A^{-1} \), or vice versa, we ensured that we end up with the identity matrix, \( I_{2} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), thus confirming the correctness of our inverse calculation.

Just remember: matrix multiplication is not commutative, meaning \( AB \) might not equal \( BA \). This is important when checking work with inverse matrices.