Question

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} -1 & 3 & 1 \\ 4 & 2 & 5 \\ -2 & 1 & 6 \end{array}\right] $$

Short answer

The determinant of the given matrix is 42.

Step-by-step solution

Understanding the Matrix

The given matrix is that of 3x3 order and the elements are \[A=\left[\begin{array}{ccc} -1 & 3 & 1 \ 4 & 2 & 5 \ -2 & 1 & 6 \end{array}\right]\]

Expanding by cofactors

To find the determinant, one way is to expand the matrix by cofactors. We will use the first row for this. The formula will look like: Det(A) = \(-1(-1)\) times the determinant of \[\left[\begin{array}{cc} 2 & 5 \ 1 & 6 \end{array}\right]\] - 3 times the determinant of \[\left[\begin{array}{cc} 4 & 5 \ -2 & 6 \end{array}\right]\] + 1 times the determinant of \[\left[\begin{array}{cc} 4 & 2 \ -2 & 1 \end{array}\right]\]

Determining the Minors

Now calculate the determinant of each 2x2 matrix. This will result in Det(A) = -12 - (-48) + 6 = 42.

Key concepts

3x3 Matrix

A 3x3 matrix is a commonly used structure in mathematics, especially in linear algebra. It consists of 3 rows and 3 columns, totaling 9 individual elements. These matrices are denoted by \[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} \] . The matrix given in the exercise is an example of a 3x3 matrix. It is represented as: \[ \begin{bmatrix} -1 & 3 & 1 \ 4 & 2 & 5 \ -2 & 1 & 6 \end{bmatrix} \] . These matrices are used to solve systems of equations, perform linear transformations, and more. Three main operations can be performed using a matrix: addition, multiplication, and finding the determinant. For a 3x3 matrix, calculating the determinant helps in understanding properties like invertibility and solving linear systems.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method to calculate the determinant of a matrix. This process involves expanding the matrix across a particular row or column that simplifies the calculations.
In our example, we expanded across the first row: <br>

• First, choose the row or column. Here, the first row is used.
• For each element, calculate its minor, which involves the determinant of the 2x2 matrix left after removing the corresponding row and column.
• Multiply each minor by the respective element in the matrix, considering the sign (-1)^(i+j), where i and j are the row and column numbers.
• The final determinant is the sum of these products.

By using cofactor expansion, the process becomes straightforward, especially if the chosen row or column contains zeros that simplify the calculation.

Graphing Utility

A graphing utility is a valuable tool in confirming mathematical results, such as the determinant of a matrix. These utilities can be software applications or online calculators that assist in executing complex operations quickly.
Most graphing utilities allow input of matrices and automatically perform calculations like finding determinants or inverses. In our exercise, using a graphing utility can verify the manual calculation of the determinant. By inputting the 3x3 matrix, you can compare the result the utility provides with your calculated value of 42.
Not only do these tools save time, but they also help reduce errors, providing a reliable method to confirm accuracy in solving mathematical problems.

Minors in Matrices

In matrix algebra, a minor is essential in calculating determinants using the cofactor expansion method. A minor is the determinant of a smaller square matrix that remains after removing one row and one column from a larger square matrix.
For example, to compute the minor for element \(-1\) in the first row of the given matrix, delete the first row and first column to obtain: \[ \begin{bmatrix} 2 & 5 \ 1 & 6 \end{bmatrix} \].
The determinant of this 2x2 matrix is the minor, calculated as \(2 \times 6 - 5 \times 1 = 7\).
Computing minors for each element in a chosen row or column is a key step in cofactor expansion. Understanding the role of minors simplifies the process of finding determinants, enabling efficient problem-solving in matrix algebra.