Problem 16 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{rrr} 2 & 7 & -3 \\ 1 & 0 & 4 \\ 4 & -1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 137.

Step 1: Calculate the Cofactor Expansion

To find the determinant of a 3x3 matrix, we'll use the method of cofactor expansion along the first row. The matrix given is: \[\begin{bmatrix} 2 & 7 & -3 \ 1 & 0 & 4 \ 4 & -1 & -2 \end{bmatrix}\]The formula for the determinant using row 1 is: \[\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\]where \(a_{ij}\) are the elements of the matrix and \(C_{ij}\) are the cofactors.

Step 2: Compute the Cofactors

Each cofactor \(C_{ij}\) is calculated as follows:- For \(C_{11}\): Remove row 1 and column 1, which gives \[\begin{vmatrix} 0 & 4 \ -1 & -2 \end{vmatrix} = (0)(-2) - (4)(-1) = 4\]- For \(C_{12}\): Remove row 1 and column 2, which gives \[\begin{vmatrix} 1 & 4 \ 4 & -2 \end{vmatrix} = (1)(-2) - (4)(4) = -18\]- For \(C_{13}\): Remove row 1 and column 3, which gives \[\begin{vmatrix} 1 & 0 \ 4 & -1 \end{vmatrix} = (1)(-1) - (0)(4) = -1\]

Step 3: Apply the Cofactor Expansion Formula

Substitute the elements of the first row and their corresponding cofactors:\[\text{det}(A) = 2 \times 4 - 7 \times (-18) - 3 \times (-1)\]Calculate each term:\[= 2 \times 4 + 7 \times 18 + 3 \times 1\]Simplify:\[= 8 + 126 + 3 = 137\]

Step 4: Interpret the Result

The determinant of the matrix is calculated to be 137. This means the matrix is invertible and the volume transformation associated with this matrix is scaled by a factor of 137.

Key Concepts

Cofactor Expansion

The cofactor expansion is a method for finding the determinant of a square matrix. When dealing with a 3x3 matrix like the one given, this method involves breaking the matrix into smaller parts, calculating their determinants, and then combining these results. In mathematical terms, you find the determinant of a 3x3 matrix by expanding along a row or column using the formula:
\[\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\]where:

• \(a_{ij}\) represents the elements of the matrix.
• \(C_{ij}\) are the cofactors, which are specific values calculated by eliminating particular rows and columns.

To find each cofactor: you remove the row and column of the element you are examining to form a smaller 2x2 determinant. Calculate this smaller determinant and multiply by \((-1)^{i+j}\) to apply the correct sign. This is the cofactor expansion method in action, allowing for a quicker calculation of the determinant on small matrices.

3x3 Matrix

A 3x3 matrix represents a mathematical object consisting of 3 rows and 3 columns. The matrix used in this exercise looks like this:
\[\begin{bmatrix}2 & 7 & -3 \1 & 0 & 4 \4 & -1 & -2 \end{bmatrix}\]Matrices are powerful tools used to represent linear transformations and systems of linear equations. A 3x3 matrix can transform or map input geometry in three-dimensional space. In our exercise, each element's placement in the matrix affects how it influences the calculation of determinant or the result of the linear transformations it represents. Three elements, one from each row and column, are used in the cofactor expansion approach to unravel the whole matrix into smaller sections that are easier to handle. This creates a layered understanding of how each part of the matrix contributes to the overall calculation, reflecting the complexity and the transformative property of matrices.

Invertible Matrix

An invertible matrix is one that has an inverse, a matrix that 'undoes' the linear transformation applied by the original matrix. For a square matrix to be invertible, a key requirement is that its determinant must not be zero.
The determinant acts like a gatekeeper:

• If the determinant is zero, the matrix is not invertible and the system of equations it represents could be dependent or inconsistent.
• If the determinant is any non-zero number, like 137 in our example, the matrix is invertible.

When you have an invertible matrix, it means reversing calculations or transformations done by the matrix is possible, restoring the original data or structure that the matrix altered. In practical applications, such as computer graphics and complex calculations, invertibility tells you that systems can be interacted with dynamically, adjusted, and reversed without loss of information.