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Chapter 2: Problem 9

Prove that if a row or a column of an \(n \times n\) matrix \(A\) consists entirely of zeros, then \(\operatorname{det}(A)=0\)

Short Answer

Expert verified
Given an \(n \times n\) matrix \(A\) with a row or a column consisting entirely of zeros, assume without loss of generality that the first row of \(A\) consists entirely of zeros. Perform row reduction to transform the matrix into an upper triangular form, say \(A'\), without changing the determinant. Since the determinant of an upper triangular matrix is the product of its diagonal elements and the first diagonal element is zero, the determinant of \(A'\) is zero. Therefore, the determinant of \(A\) is also zero.

Step by step solution

01

Set up the matrix and its determinant

Given an \(n \times n\) matrix \(A\), we assume it has a row or a column consisting entirely of zeros. If it is a row, we can set it as the first row for simplicity; if it is a column, we can consider the transpose of the matrix and work with its rows. The determinant will not be affected by taking the transpose. So, let's assume without loss of generality that the first row of \(A\) consists entirely of zeros: \(A = \begin{pmatrix} 0 & 0 & \cdots & 0 \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}\) Now, let's find the determinant of this matrix, which we will denote as \(\operatorname{det}(A)\).
02

Apply the row reduction method

To find the determinant of the matrix, we can use the row reduction method, i.e., adding a multiple of a row to another row, which does not change the determinant. We will eliminate the elements in the first column of the matrix, except for the first one which is already zero. For each row \(i\), \(2 \leq i \leq n\), we can subtract \(\frac{a_{i1}}{a_{21}}\) times the second row from the \(i\)-th row. This will transform the matrix into an upper triangular form while keeping the determinant unchanged: \(A' = \begin{pmatrix} 0 & 0 & \cdots & 0 \\ 0 & a'_{22} & \cdots & a'_{2n} \\ 0 & a'_{32} & \cdots & a'_{3n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & a'_{n2} & \cdots & a'_{nn} \end{pmatrix}\)
03

Compute the determinant of the new matrix

As the determinant of an upper triangular matrix is the product of its diagonal elements, the determinant of the new matrix \(A'\) is: \(\operatorname{det}(A') = 0 \times a'_{22} \times a'_{33} \times \cdots \times a'_{nn} = 0\)
04

Conclude the proof

Since the row reduction method we used above did not change the determinant of the matrix, we have: \(\operatorname{det}(A) = \operatorname{det}(A') = 0\) This completes the proof that if a row or a column of an \(n \times n\) matrix \(A\) consists entirely of zeros, then the determinant of \(A\) is equal to zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Upper Triangular Matrix
An upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zeros. This kind of matrix simplifies many mathematical operations, especially finding the determinant. For example, if you have a number on the diagonal of a matrix, and beneath it all zeros going downward, you have an upper triangular form.

Here鈥檚 why they are important: the determinant of an upper triangular matrix can be quickly computed as the product of its diagonal elements. This is why row operations that transform a matrix into an upper triangular form often represent the first step towards this calculation.

In practice, when you see an upper triangular matrix, you can instantly know that its determinant is calculated simply by multiplying the numbers on its diagonal together. If one of those diagonal numbers is zero, the whole product, and thus the determinant, becomes zero. This helps in simplifying your work when dealing with very complex matrices.
Row and Column Operations
Row and column operations are handy tools in matrix algebra. They allow us to change a matrix into a simpler form without altering key properties like the determinant. Common operations include switching two rows, multiplying a row by a constant, and adding or subtracting a multiple of one row from another row.

In the problem, row operations help us transform our matrix with a zero row into an upper triangular matrix. Since adding a multiple of one row to another doesn't change the determinant, we're safely simplifying the matrix without affecting its overall value. This simplification can make complex matrices more manageable and their properties easier to analyze.

It's important to know that these operations are structured so that some will change the determinant if not careful. Always remember: switching two rows flips the sign of the determinant, but the operations we use for simplification during proofs like these do not alter it.
Proof by Row Reduction
Proof by row reduction is a structured approach to showing certain properties about matrices, such as proving determinant properties. This method involves converting the matrix through allowed operations (mentioned in the previous section) until it reaches a form that's easy to analyze. Often this ultimate form is an upper triangular matrix.

In this proof, we start with a matrix having a whole row of zeros. We use row operations carefully to eliminate non-zero elements in the first column under the zero row. The idea is to get to a point where the matrix reveals its nature 鈥� that its determinant is zero due to the zero row.

Row reduction doesn鈥檛 change the determinant as long as it concerns adding or subtracting multiples of rows. Thus, what remains critical in this technique is accurately applying operations to simplify and understand the matrix without altering determinant-related properties.
Zero Row or Zero Column in Matrix
In matrix theory, a row or column full of zeroes is a red flag when calculating the determinant. Why? Because it means the matrix is singular, and its determinant is zero.

Here鈥檚 how it works: the determinant fundamentally represents scale factors in transformations, and a zero row (or column) means the transformation compresses space into a smaller dimension 鈥� effectively making it non-invertible.

If one entire row of a matrix is zero, this row contributes nothing as does its corresponding cofactor row. Thus, any operation using this matrix will involve multiplication by zero at some step, rendering the overall determinant zero.

Understanding zero rows or columns can quickly inform us about the nature of the matrix and its equations' solutions (or lack thereof), emphasizing their importance in matrix analysis.

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Most popular questions from this chapter

Evaluate each of the following determinants by inspection: (a) \(\left|\begin{array}{lll}0 & 0 & 3 \\ 0 & 4 & 1 \\ 2 & 3 & 1\end{array}\right|\) (b) \(\left|\begin{array}{rrrr}1 & 1 & 1 & 3 \\ 0 & 3 & 1 & 1 \\ 0 & 0 & 2 & 2 \\ -1 & -1 & -1 & 2\end{array}\right|\) (c) \(\left|\begin{array}{llll}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right|\)

If \(A\) is singular, what can you say about the product \(A\) adj \(A ?\)

Let \(A\) and \(B\) be \(2 \times 2\) matrices and let \\[ \begin{aligned} C &=\left(\begin{array}{ll} a_{11} & a_{12} \\ b_{21} & b_{22} \end{array}\right), \quad D=\left(\begin{array}{ll} b_{11} & b_{12} \\ a_{21} & a_{22} \end{array}\right) \\ E &=\left(\begin{array}{ll} 0 & \alpha \\ \beta & 0 \end{array}\right) \end{aligned} \\] (a) Show that \(\operatorname{det}(A+B)=\operatorname{det}(A)+\operatorname{det}(B)+\) \(\operatorname{det}(C)+\operatorname{det}(D)\) (b) Show that if \(B=E A\) then \(\operatorname{det}(A+B)=\) \(\operatorname{det}(A)+\operatorname{det}(B)\)

Let \(A\) be an \(n \times n\) matrix and \(\alpha\) a scalar. Show that \\[ \operatorname{det}(\alpha A)=\alpha^{n} \operatorname{det}(A) \\]

Let \(A\) be a \(3 \times 3\) matrix with \(a_{11}=0\) and \(a_{21} \neq 0\) Show that \(A\) is row equivalent to \(I\) if and only if \\[ \begin{array}{l} -a_{12} a_{21} a_{33}+a_{12} a_{31} a_{23} \\ +a_{13} a_{21} a_{32}-a_{13} a_{31} a_{22} \neq 0 \end{array} \\]
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