Question

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrr} -1 & 3 & 2 \\ 5 & 7 & 0 \\ -1 & 3 & 2 \end{array}\right] $$

Short answer

The determinant of the given matrix is zero because the matrix has linearly dependent rows.

Step-by-step solution

Identifying Linear Dependence

Our 3x3 matrix has the first and third rows identical: \(-1, 3, 2\). This means that they are linearly dependent.

Determinant of Linearly Dependent Rows

A matrix that has linearly dependent rows or columns always has a determinant equal to zero.

Verifying Determinant

Since we have linearly dependent rows in our matrix, the determinant of the given matrix is therefore equal to zero.

Key concepts

linear dependence

In linear algebra, linear dependence refers to the relationship between vectors in a vector space. Vectors are considered linearly dependent if at least one of the vectors can be expressed as a combination of the others. This implies that one vector does not offer any new direction or information. For example, suppose you have a group of vectors: \( \mathbf{v_1}, \mathbf{v_2}, \) and \( \mathbf{v_3} \). If there are constants \( c_1, c_2, \) and \( c_3 \), not all zero, such that:

• \( c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0} \)

then these vectors are linearly dependent.
This concept is crucial when dealing with matrices, as we explore in our next sections, because it provides insight into whether rows or columns add dimension to the space.

3x3 matrix

A 3x3 matrix is simply a square array of numbers with three rows and three columns. It is often represented as follows:\[ \begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix} \] Here, each letter represents an element of the matrix positioned at a specific row and column.

• The matrix can operate on vectors to transform them in space.
• It is often used in systems of equations, transformations, and more complex linear algebra problems.

The properties of such a matrix, like its determinant, help us understand more about the transformations it represents.

zero determinant

The determinant of a matrix is a special number that tells us about the matrix’s properties, particularly its solvability and invertibility. For a 3x3 matrix, the determinant is calculated using the rule of Sarrus or cofactor expansion, which are both methods involving multiplications and additions of its elements. A matrix with a determinant of zero is singular, meaning:

• It cannot be inverted, which implies that some solutions to matrix equations involving it may not exist or may not be unique.
• The matrix compresses space into a lower dimension, indicating all its rows or columns are linearly dependent.

Thus, finding a zero determinant directs us to check for linear dependencies among the rows or columns.

linearly dependent rows

Linearly dependent rows in a matrix indicate redundancy because their presence means one row can be written as a linear combination of others. In the context of a 3x3 matrix, when the determinant is zero, this property of linear dependence is clearly depicted.Analyzing the given matrix:\[\begin{bmatrix}-1 & 3 & 2 \5 & 7 & 0 \-1 & 3 & 2\end{bmatrix} \] Notice that the first and third rows are identical:

• This means row three is a direct linear combination of row one multiplied by 1.
• Having repeated, identical rows automatically ensures linear dependence.

Such duplications or patterns lead to a determinant of zero, confirming no new "direction" is added by having all rows. Instead, they overlap in contribution to the system.