Question

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 6 & -5 & 2 \\ 0 & 5 & -3 \\ 0 & 0 & 2 \end{array}\right] $$

Short answer

The determinant of the given matrix is 60.

Step-by-step solution

Identify the type of matrix

Recognize the given matrix as an upper triangular matrix. This means all elements below the diagonal are zero.

Apply the method for calculating the determinant of a triangular matrix

Since it's a triangular matrix, the determinant is simply the product of its diagonal elements. In this case, the diagonal elements are 6, 5, and 2.

Calculate the determinant

Multiply the diagonal elements together. \nSo, the determinant = \(6*5 *2 = 60\).

Key concepts

Upper Triangular Matrix

An upper triangular matrix is a special kind of square matrix where all the elements below the main diagonal are zero. A square matrix is any matrix with the same number of rows and columns, and the main diagonal is the one which runs from the top left to the bottom right corner.
In our exercise, the matrix given is:
\[\begin{array}{rrr}6 & -5 & 2 \0 & 5 & -3 \0 & 0 & 2\end{array}\]
This matrix is clearly upper triangular because if we look below the diagonal (the positions in rows 2 and 3 in the first column, and row 3 in the second column), we can see that all those entries are zero. These matrices are very useful because certain matrix operations, like calculation of determinants or solving matrix equations, become much simpler with this structure.

Determinant Calculation

The determinant of a matrix is a special number that can tell us things about the matrix. For an upper triangular matrix, determinant calculation is straightforward due to its structure. Usually, finding a determinant involves complex calculations including minors and cofactors, but in the case of an upper triangular matrix, all those calculations are not needed.
The method to find the determinant of an upper triangular matrix is elegantly simple: you just multiply the diagonal elements together. This is a huge simplification from the more involved process needed for other kinds of matrices. For our exercise, this property saves us time and makes finding the determinant approachable, even for those just starting to learn about matrices.

Diagonal Elements

Diagonal elements are quite possibly the heroes when it comes to upper triangular matrices. They are the elements located on the main diagonal that runs from the top left corner to the bottom right corner of a square matrix. In an upper triangular matrix, the value of the determinant is the product of these diagonal elements.
Why is this important? It is because in a determinant calculation, these are the only elements that matter in an upper triangular matrix. No matter what the other elements of the matrix are, as long as it is upper triangular, the diagonal dictates its determinant.
If we proceed to calculate the determinant of the matrix from our exercise, we'd identify these elements as 6, 5, and 2. Simply multiplying them together, as shown in the solution, yields 60, which is the determinant of the matrix. This remarkable feature significantly reduces the complexity and enhances the computational efficiency.