Problem 14 - Precalculus: Functions and Graphs

Question

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right|=-\left|\begin{array}{lll} 0 & 0 & a \\ 0 & b & 0 \\ c & 0 & 0 \end{array}\right|$$

Step-by-Step Solution

Answer

The second matrix is a permuted form of the first, changing the determinant's sign due to an odd number of swaps.

Step 1: Identify Matrix Determinant

The determinant of a diagonal matrix \( \begin{pmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{pmatrix} \) is calculated by multiplying its diagonal elements together, which gives \( a \cdot b \cdot c \).

Step 2: Identify Matrix Permutation

The second matrix \( \begin{pmatrix} 0 & 0 & a \ 0 & b & 0 \ c & 0 & 0 \end{pmatrix} \) is a permutation of the original diagonal matrix obtained by swapping rows and columns.

Step 3: Understand Effect of Permutation

Swapping columns (or equivalently rows) in a matrix changes the sign of its determinant. In this case, the permutation involves multiple swaps that results in a negative sign on the determinant.

Step 4: Apply Permutation Rule

Since the determinant of a matrix changes sign with an odd number of row or column swaps, the determinant of the second matrix becomes \( -(a \cdot b \cdot c) \), showing that it is the negative of the determinant of the original matrix.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all elements outside the main diagonal are zero. This means a diagonal matrix looks something like this:

• The matrix is square: it has the same number of rows and columns.
• The diagonal contains the values: meaning, only the top-left to bottom-right values are non-zero.
• Example: \[\begin{pmatrix} a & 0 & 0 \0 & b & 0 \0 & 0 & c \\end{pmatrix}\]

Calculating the determinant of a diagonal matrix is straightforward: simply multiply its diagonal elements together. In our example, the determinant is \(a \cdot b \cdot c\). This simplicity is one reason why diagonal matrices are frequently encountered and quite useful in linear algebra.

Matrix Permutation

To understand matrix permutation, think of it as rearranging the rows and columns of a matrix. This might remind you of a puzzle where the pieces need to be moved to form a new pattern.

• Permutation changes the order of elements in the matrix.
• It can involve swapping two or more rows or columns.
• The effect is similar to shuffling cards but with specific mathematical rules.

In our exercise, we see a permutation as the rows and columns are swapped from one matrix to another:\[\begin{pmatrix} 0 & 0 & a \0 & b & 0 \c & 0 & 0 \\end{pmatrix}\]While the numbers remain the same, their positions are different. This raises an expectation of transformation in the determinant as well.

Determinant Sign Change

When you change the arrangement of rows or columns in a matrix through permutation, it directly affects the determinant's sign. Permutations can be even or odd, depending on the number of swaps you've made.

• Each swap of two rows or two columns inverts the determinant's sign.
• An even number of swaps will restore the original sign.
• An odd number of swaps results in a sign change. \[- (a \cdot b \cdot c)\]

In our particular example, after rearranging a diagonal matrix, the resultant matrix undergoes enough swaps to yield a negative determinant. This alteration in sign is a key aspect of understanding permutations' role in linear algebra.

Row and Column Swaps

Row and column swaps within matrices are fundamental operations in linear algebra. Knowing how they impact the determinant is crucial for a deeper understanding of matrices.

• Swapping two rows or two columns is akin to rearranging the order of operations.
• These swaps are the core mechanism behind permutations.
• It's essential to note that each individual swap will change the determinant's sign; one swap results in multiplying the determinant by \(-1\).

Consider our original diagonal matrix:\[\begin{pmatrix} a & 0 & 0 \0 & b & 0 \0 & 0 & c \\end{pmatrix}\]By performing swaps, the new configuration results in:\[\begin{pmatrix} 0 & 0 & a \0 & b & 0 \c & 0 & 0 \\end{pmatrix}\]The process illustrates how a few swaps between rows and columns efficiently result in the negative determinant of the original matrix.