Question

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrr} 3 & 2 & -1 \\ -6 & -4 & 2 \\ 5 & -7 & 9 \end{array}\right] $$

Short answer

The determinant of the given matrix equals to zero, since after applying row operations, a zero row was formed in the matrix. According to the properties of determinants, a determinant of a matrix with a zero row is equal to zero.

Step-by-step solution

Apply row operations

First, use row operations to simplify the matrix. A multiplied row can be added to another row without changing the determinant. Subtract 2 times the first row from the second row. The resulting matrix becomes: \[\left[\begin{array}{rrr} 3 & 2 & -1 \\ 0 & 0 & 0 \\ 5 & -7 & 9 \end{array}\right]\]

Calculate the determinant

With zero row in the matrix, this greatly simplifies our computation of the determinant. For a 3x3 matrix, the determinant can be calculated using the rule of Sarrus. But since there are zero rows, the determinant is zero.

Key concepts

Matrix Row Operations

Matrix row operations are fundamental transformations that can be applied to matrices.
They help simplify matrices without changing their essential properties, such as their determinant.
These operations, which include row addition, row subtraction, and row multiplication, are crucial in solving systems of equations and finding determinants efficiently.

• Addition/Subtraction: You can add or subtract one row from another. This is like combining equations in algebra.
• Scalar Multiplication: You may multiply a row by a non-zero scalar (a constant). This stretches or shrinks the row entries without changing the row's essential effect.
• Row Interchange: Swapping two rows changes the determinant's sign but doesn't alter its absolute value.

In the provided exercise, we used row subtraction to eliminate numbers and create a row of zeroes.
Specifically, subtracting 2 times the first row from the second row yielded a new matrix. This result showed a new row, consisting entirely of zeroes, which drastically simplifies determining the matrix's determinant.

3x3 Matrices

A 3x3 matrix is a square matrix with three rows and three columns.
This type of matrix is frequently encountered in various applications ranging from physics to computer graphics.
A matrix of this size needs specific methods, like the Rule of Sarrus, to compute its determinant effectively.
Understanding the layout and structure of 3x3 matrices is essential to performing calculations with them.A 3x3 matrix generally takes the form:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]The determinant of a 3x3 matrix is crucial because it exposes whether the matrix is invertible.
If the determinant is zero, the matrix is singular, implying it has no inverse.
In our exercise, after simplifying the rows, achieving a row of zeroes immediately concluded the determinant was zero, indicating the matrix's singularity.

Rule of Sarrus

The Rule of Sarrus is a quick, intuitive method to find the determinant of a 3x3 matrix.
It involves a process similar to taking cross-products in vector operations.
This rule is particularly beneficial when direct substitutions or calculations are cumbersome.
The steps for applying the Rule of Sarrus are as follows:

• Write down the matrix, and repeat the first and second columns to the right.
• Find the sum of the products of the diagonals going from the upper left to the lower right.
• Subtract the sum of the products of the diagonals going from the lower left to the upper right.

Mathematically, for a matrix \( A \), the determinant is:\[\text{det}(A) = aei + bfg + cdh - ceg - bdi - afh\]In the context of our exercise, though we started using the Rule of Sarrus, the presence of a zero row simplified the process.
A zero row in a 3x3 matrix makes the determinant calculation straightforward: the determinant is automatically zero.