Question

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 5 & -3 & 2 \\ 7 & 5 & -7 \\ 0 & 6 & -1 \end{array}\right] $$

Short answer

The determinant of the given matrix is 248.

Step-by-step solution

Identify the Matrix Elements

The given matrix is \[ \begin{bmatrix} 5 & -3 & 2 \ 7 & 5 & -7 \ 0 & 6 & -1\end{bmatrix} \]. Its elements are a11 = 5, a12 = -3, a13 = 2, a21 = 7, a22 = 5, a23 = -7, a31 = 0, a32 = 6, a33 = -1.

Calculate the minor matrices

To proceed with the cofactor expansion, we need to calculate the minor matrices for the elements of the first row.For a11 (5), delete the first row and the first column: \[\begin{bmatrix}5 & -7\6 & -1\end{bmatrix}\].For a12 (-3), delete the first row and the second column: \[\begin{bmatrix}7 & -7 \ 0 & -1\end{bmatrix}\].For a13 (2), delete the first row and the third column: \[\begin{bmatrix}7 & 5 \ 0 & 6\end{bmatrix}\].

Compute the determinants of the minor matrices

For the minor matrix of a11: \((5*-1) - (-7*6) = -5 + 42 = 37\)For the minor matrix of a12: \((7*-1) - (-7*0) = -7\)For the minor matrix of a13: \((7*6) - (5*0) = 42\)

Cofactor Expansion

The determinant of the original matrix is calculated by summing up the products of the elements of the first row and their corresponding minors, with an alternating sign. Therefore, Determinant = \(a11 * Minor(a11) - a12 * Minor(a12) + a13 * Minor(a13)\)= \(5*37 - -3*-7 + 2*42 = 185 - 21 + 84 = 248\).

Key concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method for calculating the determinant of a square matrix. It involves breaking down a large matrix into smaller, more manageable pieces called 'minors', and then using these pieces to find the determinant in a step-by-step approach.

The determinant is found by taking the elements of any row or column, multiplying each element by its corresponding 'cofactor', and summing these products. The cofactor of an element is the determinant of the minor associated with that element, multiplied by -1 raised to the power of the sum of the element's row and column indices. This sign alternation is crucial and reflects the checkerboard pattern of signs in the cofactor matrix. For instance, the cofactor of an element in the first row and first column is the determinant of the minor without the sign change, while the cofactor of an element in the first row and second column includes a sign change from positive to negative.

An example from the original exercise, using cofactor expansion along the first row, would be the following:

• Calculate the minor of each element in the first row.
• Apply the alternating signs starting with positive for the top-left corner.
• Multiply each element by its corresponding cofactor.
• Sum up these products to get the determinant of the matrix.

The actual calculation would then be a step-wise application of the above procedure to eventually arrive at the determinant.

Matrix Minors

The concept of a 'matrix minor' is central to understanding the cofactor expansion technique. A minor of a matrix at a particular position is the determinant of a smaller matrix, created by removing the row and column of the element in question. This smaller matrix must be square, and its determinant gives the minor for the element in the original matrix.

To compute a minor, you perform the following steps:

• Omit the row and column of the element for which you're finding the minor.
• Calculate the determinant of the resulting smaller matrix.

This process is repeated for various elements in the matrix, particularly when using the cofactor expansion method of finding determinants. In our exercise scenario, the minors were computed for the first row's elements, which enabled the solution to progress with the cofactor expansion. The minors provided the necessary scalar values that, along with their corresponding cofactors, contributed to the final determinant computation.

Matrix Elements

At the heart of any matrix are its elements, which are the individual numbers or expressions contained within the rows and columns of the matrix. Understanding a matrix's elements is fundamental to performing any matrix operation, including finding the determinant through cofactor expansion.

In mathematical notation, the elements of a matrix are denoted as 'a[ij]', where 'i' represents the row number and 'j' represents the column number. For the matrix given in the exercise, we have nine matrix elements with labels such as 'a11' for the element in the first row and first column, 'a12' for the first row and second column, and so on.

These elements serve as the foundation for calculating minors and cofactors, since each minor is associated with one matrix element. When conducting a cofactor expansion, it's crucial to compute the determinants of the minors for the matrix elements selected for expansion. By meticulously aligning each matrix element with its minor and respective cofactor, one can accurately carry out the procedures to find the overall determinant of the matrix.