Problem 45 - Precalculus with Limits

Question

In Exercises 39-54, find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. \(\left[ \begin{array}{r} 1 & 4 & -2 \\ 3 & 2 & 0 \\ -1 & 4 & 3 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is -2

Step 1: Initial matrix

Given the initial matrix: \( \left[ \begin{array}{r} 1 & 4 & -2 \ 3 & 2 & 0 \ -1 & 4 & 3 \end{array} \right] \)

Step 2: Expansion by cofactors

Choose the first row to expand by cofactors, as it has the simplest numbers. The determinant of a 3x3 matrix can be calculated using the Rule of Sarrus by subtracting the sum of the products of the diagonals going up from the sum of the products of the diagonals going down. The determinant will be \( det(A) = 1*(2*3 - 0*4) - 4*(3*3 - 0*-1) - 2*(3*4 - 2*-1) \)

Step 3: Simplification and calculation

Simplify and compute the above computation to find the determinant. det(A) = 1*(6 - 0) - 4*(9 - 0) - 2*(12 - (-2)) = 6 - 36 + 28 = -2

Key Concepts

Matrix expansion

Matrix expansion, especially for 3x3 matrices, involves breaking down the matrix into smaller operations that make it easier to find determinantal values. This approach leverages the smaller components to build up to the determinant in an organized way. For a 3x3 matrix:\[\left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right]\]The determinant can be computed through expansion methods, such as the Rule of Sarrus or cofactor expansion. These techniques involve calculating the influences of each element in the matrix separately, rather than trying to solving all elements at once. Matrix expansion allows a complex polynomial expression to be broken down into simpler calculations, which is key to determining the matrix determinant.

Rule of Sarrus

The Rule of Sarrus is particularly handy for computing the determinant of a 3x3 matrix. It relies on a visualization of diagonal products. This method is only applicable to 3x3 matrices:Imagine extending the first two columns of the matrix to the right of the matrix. Calculate the products of the diagonals slanting downward and upward.

• Downward diagonals: Multiply the elements diagonally from left to right, going down: \(a \times e \times i, b \times f \times g, c \times d \times h\).
• Upward diagonals: Do the same, but from right to left, going up: \(c \times e \times g, a \times f \times h, b \times d \times i\).

Subtract the sum of the upward diagonals from the sum of the downward diagonals. This gives the determinant of the matrix. While the Rule of Sarrus is very visual and straightforward for small matrices, it cannot be used for larger ones, where cofactor expansion becomes essential.

Cofactor expansion

Cofactor expansion, or Laplace expansion, is a powerful method to find the determinant, especially in larger matrices. It involves choosing any row or column and expanding the determinant calculation by each element multiplied by its corresponding cofactor. A cofactor is defined as \((-1)^{i+j}\times\) minor of the element, where \(i\) and \(j\) denote the element's position. For the given matrix, the simplest choice was the first row. The calculation involves:

• For the element at \((1,1)\) position: \(1\times\) the determinant of the submatrix formed by removing the first row and column.
• For the element at \((1,2)\): \(-4\times\) the determinant of its submatrix.
• For the element at \((1,3)\): \(-2\times\) the determinant of its submatrix.

Solving these sub-determinants and weighing them according to their cofactors breaks down complex determinant calculations, allowing them to be manageable even for larger matrices!