Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{rr} -7 & 33 \\ 4 & -19 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the matrix is \(\left[\begin{array}{cc} -19/det & -33/det \\ -4/det & -7/det \end{array}\right]\) where 'det' is the determinant of the original matrix.

Step 1: Calculate the determinant

Calculate the determinant (denoted as 'det') of the given matrix. det = (-7 * -19) - (33 * 4)

Step 2: Check if the inverse exists

Check if the determinant is not equal to 0. If it's zero, the inverse of the matrix does not exist.

Step 3: Calculate the inverse

If the determinant is not 0, the inverse can be calculated as follows: swap the elements in the main diagonal, multiply the other two elements by -1, and then divide all elements by the determinant. The resulting matrix will be the inverse.

Step 4: Simplify the inverse matrix

Perform the calculations and simplify the resulting matrix. This will provide the final inverse.

Key Concepts

Determinant Calculation

Determining the determinant of a matrix is the first step in finding its inverse. The determinant gives us crucial information about the matrix properties. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det \) is calculated as:\[\det = (a \times d) - (b \times c)\]In our given matrix, \( \begin{bmatrix} -7 & 33 \ 4 & -19 \end{bmatrix} \), calculate the determinant step by step:

• Multiply the top left and bottom right elements: \(-7 \times -19 = 133 \)
• Multiply the top right and bottom left elements: \(33 \times 4 = 132 \)
• Subtract these results: \(133 - 132 = 1\)

With a determinant of 1, we can proceed to calculate the inverse since the determinant is not zero.

Inverse Matrix Formula

The inverse of a matrix \( A \) is denoted as \( A^{-1} \), and it only exists if the determinant is not zero. The general formula to find the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is:\[A^{-1} = \frac{1}{\det(A)} \times \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]For our matrix \( \begin{bmatrix} -7 & 33 \ 4 & -19 \end{bmatrix} \),

• Swap positions of the diagonal elements: \(-7\) and \(-19\) become \(-19\) and \(-7\)
• Change signs of the off-diagonal elements: 33 and 4 become -33 and -4
• Divide each element by the determinant (1): \[A^{-1} = \begin{bmatrix} -19 & -33 \ -4 & -7 \end{bmatrix} \]

Since the determinant is 1, the calculation is straightforward, and the simplified inverse is the same as it appears.

Matrix Algebra

Matrix algebra involves operations like addition, subtraction, multiplication, and finding inverses. Understanding these operations is crucial for manipulating matrices effectively. Here's a brief overview:

• Addition: Add corresponding elements in two matrices of the same size.
• Subtraction: Subtract corresponding elements in two matrices of the same size.
• Multiplication: Multiply rows by columns — product matrices only exist under specific dimension conditions.
• Inverse: Exists only if the determinant isn’t zero; it's pivotal for solving linear equations.

In our specific example, the inverse calculation is a critical component of matrix algebra, allowing for transformations and solving systems where the matrix is used.

Linear Algebra Concepts

Linear algebra is a branch of mathematics concerned with vector spaces and linear transformations. Key concepts include:

• Vectors: Building blocks of linear algebra, representing magnitude and direction.
• Matrices: Rectangular arrays used to represent linear transformations and systems of equations.
• Determinants: Scalar values that can indicate non-invertibility of a matrix.
• Inverses: Used to find solutions to matrix equations or reverse transformations.

These concepts are intertwined. For example, finding a matrix's inverse requires an understanding of determinants. An inverse allows the solving of equations like \( A\vec{x} = \vec{b} \), where \( \vec{x} = A^{-1}\vec{b} \) when \( A \) is invertible, highlighting its importance in linear algebra's application.