Problem 16 - 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 \(A^{-1} A=I_{2}\) $$ A=\left[\begin{array}{ll} {2} & {-6} \\ {1} & {-2} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix \( A = \[ \[2, -6\], \[1, -2\] \] \) is \( A^{-1} = \[ \[-0.2, 0.6\], \[-0.1, 0.2\] \] \).

Step 1: Apply the formula

The matrix \( A = \[ \[2, -6\], \[1, -2\] \] \). According to the formula, first calculate the denominator, which is ad-bc. In this case, a=2, b=-6, c=1 and d=-2. Hence, ad-bc=(2*-2)-(-6*1)=4+6=10. The denominator is not equal to zero, hence an inverse exists. Apply the formula to find the inverse matrix. This results in \( A^{-1} = 1/10 \[ \[-2, 6\], \[-1, 2\] \] = \[ \[-0.2, 0.6\], \[-0.1, 0.2\] \] \).

Step 2: Verify the inverse matrix

Multiply the original matrix A with its inverse. If the resulting matrix is the identity matrix \( I_2 = \[ \[1, 0\], \[0, 1\] \] \), then the inverse matrix is correct. Starting with \( A A^{-1} \), multiply each element with corresponding element in the inverse matrix and sum the products:\( A A^{-1}= \[ \[2, -6\], \[1, -2\] \] * \[ \[-0.2, 0.6\], \[-0.1, 0.2\] \] = \[ \[1, 0\], \[0, 1\] \] \).Next, multiply \( A^{-1}A \) to further validate the inverse:\( A^{-1}A = \[ \[-0.2, 0.6\], \[-0.1, 0.2\] \] * \[ \[2, -6\], \[1, -2\] \] = \[ \[1, 0\], \[0, 1\] \] \).In both cases, the result is the identity matrix, validating the calculated inverse.

Key Concepts

2x2 matrix

A 2x2 matrix is a simple, yet powerful, mathematical tool used widely in linear algebra. It consists of two rows and two columns and can be expressed as:

• \( \boldsymbol{A} = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \).

Here, \( a, b, c, \) and \( d \) are the elements of the matrix. Each element can be a real number, a variable, or even another expression. A 2x2 matrix can represent various transformations and operations, such as rotating or scaling objects in geometry. This matrix can also be used for solving systems of linear equations, making it a critical component of mathematical studies.
Understanding the layout and functionality of a 2x2 matrix is crucial because it forms the basis for more complex matrices and operations.

inverse matrix verification

Once an inverse matrix is calculated, checking its accuracy is crucial. The verification involves ensuring that multiplying the matrix by its inverse yields the identity matrix. The identity matrix for a 2x2 array is:

• \( I_2 = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \)

For a matrix \( \boldsymbol{A} \) and its inverse \( \boldsymbol{A}^{-1} \) to be correct, the following conditions must hold:

• \( A \times A^{-1} = I_2 \)
• \( A^{-1} \times A = I_2 \)

Each product must individually return the identity matrix, confirming the accuracy of the inverse. In our example, both operations returned:

• \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \)

This corroborates that the inverse calculation was performed correctly. The step ensures that any actions relying on the inverse matrix, such as solving equations or transformations, are based on accurate computations.

determinant calculation

Calculating the determinant is an essential step in identifying whether a matrix can have an inverse. The determinant of a 2x2 matrix \( \boldsymbol{A} = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is given by:

• \( \,det(A) = ad - bc \)

If the determinant is non-zero, the matrix is invertible. In our case, for the matrix \( \left[ \begin{array}{cc} 2 & -6 \ 1 & -2 \end{array} \right] \), the calculation was:

• \( 2 \times (-2) - (-6) \times 1 = -4 + 6 = 2 \)

The resulting determinant value of 2 confirms the matrix is invertible. Should the determinant equal zero, it indicates that the matrix cannot be inverted. Therefore, ensuring the determinant is calculated correctly is vital for the correctness of subsequent operations. Thorough determinant calculation helps avoid incorrect assumptions about matrix invertibility and reliability in calculations.