Question

If \(A\) has three columns with only the top two entries nonzero, show that \(\operatorname{det} A=0\).

Short answer

The determinant of matrix A is zero because it has an entire row of zeros.

Step-by-step solution

Identify Matrix Structure

Matrix A has three columns, but each column has its last entry as zero. Let's assume it is a 3x3 matrix:\[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ 0 & 0 & 0 \end{bmatrix}.\] The third row being zero is the key to solving this problem.

Understand Determinant Properties

Recall that the determinant of a 3x3 matrix is given by:\[\operatorname{det} A = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}).\] Since the third row is all zeros, each term involving any element from the third row becomes zero.

Calculate the Determinant

Substitute the zero elements from the third row into the general formula for the determinant:\[\operatorname{det} A = a_{11}(a_{22} \times 0 - a_{23} \times 0) - a_{12}(a_{21} \times 0 - a_{23} \times 0) + a_{13}(a_{21} \times 0 - a_{22} \times 0).\]Simplifying further, each term results in multiplication by zero, hence:\[\operatorname{det} A = 0.\]

Conclusion and Result Verification

From the calculation and determinant properties, we've confirmed that whenever a row (or column) consists entirely of zeros, the determinant is zero. Therefore, the determinant of this matrix A is zero: \(\operatorname{det} A = 0\).

Key concepts

Matrix Structure

Matrices are structured arrays of numbers that represent data or linear transformations. In this problem, we are dealing with a 3x3 matrix, which is a square matrix with three rows and three columns. Each element of the matrix is identified by two indices: the row and the column.
This matrix has a special structure where all entries in its third row are zeros:\[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ 0 & 0 & 0 \end{bmatrix}\].
Understanding the matrix structure is crucial because specific properties or conditions in the matrix, such as the presence of a zero row, directly affect calculations like determinants.

• A zero row means all its elements are zero.
• Matrices like this often arise in problems involving transformations or systems with certain constraints.
• The matrix's structure can significantly simplify or dictate the method of solving problems related to determinants and linear algebra operations.

Zero Row Determinant

A zero row in a matrix means that every entry in that row is zero. This has a direct and simplifying impact on calculating the determinant.
When calculating the determinant of a 3x3 matrix, several products and subtractions are involved, as in the expression:\[\operatorname{det} A = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}).\]
However, if any row or column of the matrix is filled entirely with zeros, each term of the determinant calculation that uses these zero elements results in zero.

• Since the third row is zero, any product involving \(a_{31}, a_{32} \text{ or } a_{33}\) becomes zero.
• This results in the entire determinant evaluating to zero, regardless of the other entries in the matrix.
• Therefore, knowing a matrix has a zero row allows for quick determinant computation: \(\operatorname{det} A = 0\).

Properties of Determinants

Determinants hold several properties that make computations in linear algebra manageable and offer insights into matrix characteristics.
One crucial property is that the determinant of a matrix with an entire row or column of zeros is always zero. This is because the determinant can be thought of as a volume scaler or transformation factor.

• A zero determinant means the transformation collapses the volume into a lower dimension, often flattening it entirely to zero.
• Other properties include linearity, where determinant values change predictably with row additions or scalar multiplications.
• Determinants also help ascertain invertibility: a non-zero determinant signifies an invertible matrix, while a zero determinant does not.

Understanding these properties helps in quickly identifying scenarios—like a zero row matrix—where calculations can be simplified or further interpreted in practical terms related to linear transformations.