Question

Show that det \(A=0\) if \(A\) has a row or column consisting of zeros.

Short answer

The determinant \(\text{det}(A)\) is 0 if \(A\) has a row or column of zeros.

Step-by-step solution

Understand the Determinant Properties

The determinant of a matrix, specifically an \(n\times n\) square matrix \(A\), is a scalar value that is a function of the entries of the matrix. One of the key properties of determinants is that if any row or column of the matrix consists entirely of zeros, the determinant of that matrix is zero.

Set Up the Matrix with a Zero Row

Consider an \(n\times n\) matrix \(A\) where one of the rows, say the \(i\)-th row, consists entirely of zeros: \[A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ 0 & 0 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}\]

Calculate the Determinant Using Cofactor Expansion

We can use cofactor expansion along the zero row to calculate the determinant of \(A\). The determinant is given by the sum of the products of the elements of that row and their corresponding cofactors.

Apply Cofactor Calculation

For the zero row, each element is zero, that is \(a_{i1} = a_{i2} = \cdots = a_{in} = 0\). Therefore:\[\text{det}(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in} = 0 \cdot C_{i1} + 0 \cdot C_{i2} + \cdots + 0 \cdot C_{in} = 0\]Each term in the sum is zero because each \(a_{ij} = 0\), and this results in the determinant being zero.

Conclusion

Since the determinant is calculated as zero when a row of zeros is present, we conclude that the matrix \(A\) with a zero row indeed has a determinant of zero. This property also holds for columns by a similar argument.

Key concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. It's like opening a doorway into understanding how determinants behave, especially in larger matrices. When performing cofactor expansion, you break down a matrix into smaller parts, which are easier to handle.
Usually, you choose a specific row or column to expand along, and this choice often depends on where zeros appear, simplifying the calculation greatly.

A determinant of a matrix is the sum of the products of the elements of a row (or column) and their corresponding cofactors. Cofactors themselves are calculated by finding the determinant of smaller matrices, specifically those formed by removing the row and column of an element. This process can be repeated on these smaller matrices. It’s a recursive technique. Understanding this helps in directly explaining why a row or column of zeros leads to a determinant of zero—it multiplies to nothing!

• If you choose a row with all zeros for cofactor expansion, all terms in the expansion will be zero, as each element in that row is multiplied by its cofactor, but every element is zero.
• This makes calculations much more manageable and is a precise, accepted mathematical approach for finding determinants, particularly in theoretical proofs.

Zero Row or Column

The rule about having a row or column of zeros in a matrix leading to a zero determinant is a very handy shortcut. It saves time and prevents unnecessary calculations. Knowing and applying this property quickly tells you a lot about the matrix without going through complex computations.

For instance, when one entire row or column is zeros in a square matrix, the contribution to the determinant from that row or column is nil. As each element in that row or column is zero, multiplying them with anything else, like their cofactors, results in zero. The determinant ultimately depends on summing these products, so if every product is zero, the determinant is zero.

• This holds equally true whether it's a row full of zeros or a column. The position of the zeros doesn't matter when the entire row or column is zero.
• This property simplifies decision-making when choosing which row or column to utilize for cofactor expansion, making this a powerful strategy in larger systems.

Matrix Properties

Matrices possess a variety of properties that are crucial when determining the behavior of their determinants. Understanding these properties helps one grasp why, for instance, a zero row or column impacts the determinant so dramatically.

A few notable properties of determinants include:

• Linearity: The determinant behaves linearly with respect to the rows and columns of the matrix. This means that if you add a multiple of one row to another, the determinant remains the same.
• Zero Rows/Columns: As discussed, if any row or column is entirely zero, the determinant equals zero. This is because no non-zero products will come out of the cofactor expansions.
• Row Interchange: Switching two rows or columns in a matrix results in the determinant switching sign. However, this doesn't impact the zero determinant rule.
• Diagonal Rule: In triangular matrices (either upper or lower), the determinant equals the product of the diagonal elements. If one of these is zero, then so is the determinant.

Understanding these principles not only helps with problem solving but also with recognizing patterns and connections between various algebraic concepts. Each property serves as a tool in showcasing how determinants can be manipulated or simplified into more digestible pieces.