Question

Find the determinant of the given matrix using cofactor expansion along the first row. 13\. \(\left[\begin{array}{ccc}3 & 2 & 3 \\ -6 & 1 & -10 \\ -8 & -9 & -9\end{array}\right]\) 14\. \(\left[\begin{array}{ccc}8 & -9 & -2 \\ -9 & 9 & -7 \\ 5 & -1 & 9\end{array}\right]\) 15\. \(\left[\begin{array}{ccc}-4 & 3 & -4 \\ -4 & -5 & 3 \\ 3 & -4 & 5\end{array}\right]\) 16\. \(\left[\begin{array}{ccc}1 & -2 & 1 \\ 5 & 5 & 4 \\ 4 & 0 & 0\end{array}\right]\) 17\. \(\left[\begin{array}{ccc}1 & -4 & 1 \\ 0 & 3 & 0 \\ 1 & 2 & 2\end{array}\right]\) 18\. \(\left[\begin{array}{ccc}3 & -1 & 0 \\ -3 & 0 & -4 \\ 0 & -1 & -4\end{array}\right]\) 19\. \(\left[\begin{array}{ccc}-5 & 0 & -4 \\ 2 & 4 & -1 \\ -5 & 0 & -4\end{array}\right]\) 20\. \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]\)

Short answer

The determinant of matrix 13 is -59.

Step-by-step solution

Identify the First Row Elements

Consider matrix 13: \( \begin{bmatrix} 3 & 2 & 3 \ -6 & 1 & -10 \ -8 & -9 & -9 \end{bmatrix} \). Our task is to find the determinant using cofactor expansion along the first row, where the elements are 3, 2, and 3.

Calculate the Minors for Each Element

For the element 3 in the first row, exclude its row and column to get the submatrix \( \begin{bmatrix} 1 & -10 \ -9 & -9 \end{bmatrix} \). For the element 2, the submatrix is \( \begin{bmatrix} -6 & -10 \ -8 & -9 \end{bmatrix} \). For the third element 3, the submatrix is \( \begin{bmatrix} -6 & 1 \ -8 & -9 \end{bmatrix} \).

Compute Determinants of Minors

Calculate the determinants of each minor:- Det(\( \begin{bmatrix} 1 & -10 \ -9 & -9 \end{bmatrix} \)) = \((1)(-9) - (-10)(-9) = -9 - 90 = -99\).- Det(\( \begin{bmatrix} -6 & -10 \ -8 & -9 \end{bmatrix} \)) = \((-6)(-9) - (-10)(-8) = 54 - 80 = -26\).- Det(\( \begin{bmatrix} -6 & 1 \ -8 & -9 \end{bmatrix} \)) = \((-6)(-9) - (1)(-8) = 54 + 8 = 62\).

Apply Cofactor Signs and Sum

Assign the appropriate cofactor sign to each element from the first row: +, -, +. Calculate:- First element: \(+3 \times -99 = -297\).- Second element: \(-2 \times -26 = 52\).- Third element: \(+3 \times 62 = 186\).

Compute the Determinant

Add the results of the cofactor calculations: \(-297 + 52 + 186 = -59\). This sum gives the determinant of the matrix.

Key concepts

Cofactor Expansion

Cofactor expansion is a method used to determine the determinant of a matrix, especially useful for larger matrices like a 3x3. This technique involves breaking down the matrix into smaller parts, making computations manageable. To perform a cofactor expansion, start by selecting any row or column. In our case, we use the first row for expansion.
Each element in this row will be the focus of its own calculation.

• For each element, compute its cofactor, which entails multiplying the element by the determinant of its corresponding minor.
• Alternate the signs starting with a positive sign for the first element, negative for the next, and so on.
• Combine all these results to get the determinant of the matrix.

This approach allows for flexibility; any row or column can be used, although sometimes one row or column might make calculations simpler due to zero entries. Using cofactor expansion demonstrates a straightforward method for slicing the problem of finding determinants into smaller, more approachable pieces.

Minors

Minors are essential components when working with cofactor expansions. The minor of an element in a matrix is the determinant of a smaller matrix, formed by deleting the row and column of that element.
For example, in the matrix from the exercise, when determining the minor of the first element of the first row, you eliminate both the first row and the first column.

• This leaves you with a submatrix. In our situation, this would be \[ \begin{bmatrix} 1 & -10 \ -9 & -9 \end{bmatrix} \] for the first element, which is 3.

Calculating the determinant of a 2x2 submatrix is straightforward: multiply the top-left and bottom-right entries, subtract the product of the top-right and bottom-left entries.
This minor calculation is repeated for each element in the selected row or column. Minors are what make cofactor expansion possible, providing the smaller determinants that factor into the final determinant calculation.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns, containing nine elements. Such matrices are commonly encountered in linear algebra and have applications across numerous fields, from physics to computer science. The determinant of a 3x3 matrix can give insights, such as whether a matrix is invertible.
Understanding the layout is critical when performing calculations like cofactor expansion:

• Each element can represent an entry that affects calculations in rows and columns.
• Align the matrix into visual blocks or elements to simplify the process of deletion when finding minors.

The compact size of a 3x3 matrix allows for hand computations of its determinant, making it an excellent candidate for learning the fundamentals of matrix operations. Mastery over these basics, like evaluating determinants, is foundational for handling larger matrices and more complex linear algebra problems.

Determinant Calculation

Determinant calculation involves a series of steps leading to a single scalar number, offering valuable information about a matrix. In a 3x3 matrix, the determinant can be found using various methods, but cofactor expansion along a row is a primary technique.

• Begin with selecting a specific row for expansion; often, the first row is chosen for simplicity as seen in the example.
• Determine the minors associated with each element in the row.
• Compute the cofactor for each, factoring in alternating signs (+, -, + for the first row).

Finally, sum these signed values to arrive at the determinant. In the case of the given matrix, these calculations sum up to -59.
It's essential to recognize the determinant not just as a calculated number but as a tool. It helps to determine matrix properties, like invertibility or the volume scaling factor of linear transformations depicted by the matrix. Understanding these implications propels one ahead in mastering linear transformations and matrix theory.