Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{cccc}5 & -5 & 0 & 1 \\ 2 & 4 & -1 & -1 \\ 5 & 0 & 0 & 4 \\\ -1 & -2 & 0 & 5\end{array}\right]\)

Short answer

The determinant of the matrix is 80.

Step-by-step solution

Choose a Row for Expansion

Let's choose the first row of the matrix for the cofactor expansion. This row contains elements: 5, -5, 0, and 1, which will simplify some calculations because one of the entries is zero.

Calculate the Cofactors

For each element in the chosen row, we need to compute the cofactor. The cofactor is calculated as the determinant of the submatrix obtained by removing the row and column of the element. Also, multiply the determinant of the submatrix by \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number of the element.

Evaluate Minor for Element 5

The submatrix after removing the first row and first column is \([\begin{smallmatrix}4 & -1 & -1 \ 0 & 0 & 4 \ -2 & 0 & 5 \end{smallmatrix}]\). We calculate its determinant.

Calculate Determinant of 3x3 Matrix (Minor for 5)

Using the first row to expand this 3x3 minor: \(4 \cdot ((0\cdot5)-(4\cdot0)) - (-1)\cdot((0\cdot5)-(4\cdot(-2))) - (-1)\cdot((0\cdot0)-((-2)\cdot0))\). This simplifies to \(4\cdot0 + 2\cdot4\cdot1 + 0 = 8\).

Evaluate Minor for Element -5 and 1

The element 0 has a zero cofactor because multiplying anything by zero yields zero.Evaluate the minor obtained from \((-5)\) and \(1)\) in a similar manner and sum all results.

Evaluate Remaining Minors and Cofactors

For element -5 in the first row and column 2, we remove row 1 and column 2, forming the submatrix \([\begin{smallmatrix}2 & -1 & -1 \ 5 & 0 & 4 \ -1 & 0 & 5 \end{smallmatrix}]\) and calculate its determinant using cofactor expansion. Similarly, calculate for element 1.

Compute Final Determinant

Sum the products of each element from the first row with their corresponding cofactors. For element 1 in column 4, calculate the 3x3 determinant of the eliminated submatrix. Add these to find the matrix's determinant.

Key concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. It involves the expansion along a specific row or column. Each element in the chosen row or column is multiplied by its cofactor. This leads to an expression for the determinant.

• Choose a row or column for expansion.
• For each element, compute the minor: the determinant of the submatrix after removing the element's row and column.
• Calculate the cofactor, which includes multiplying the minor by \((-1)^{i+j}\).
• Sum these products to obtain the final determinant.

This process simplifies when zeros are present in the chosen row or column, as their cofactors contribute nothing to the sum.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and operations on them. It's foundational in many disciplines, including physics and computer science. Understanding how to manipulate matrices is crucial.

• Addition and subtraction involve element-wise operations.
• Matrix multiplication is more complex, involving the dot product of rows and columns.
• The inverse of a matrix, used in solving matrix equations, is related to determinants.
• The determinant gives insights into the matrix's properties, like whether it's invertible.

Applying these operations can solve linear equations, perform transformations, and even contribute to data analysis.

3x3 Matrix Determinant

Calculating a 3x3 matrix determinant is an essential skill in matrix algebra. For a matrix \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]you calculate the determinant using:\[D = a(ei - fh) - b(di - fg) + c(dh - eg)\]Here's why it's crucial:

• It determines if a matrix is invertible (a non-zero determinant means it's invertible).
• It aids in understanding linear transformations represented by the matrix.
• It's a building block in calculating determinants for larger matrices through cofactor expansion.

This calculation is straightforward, crucial for solving larger systems using cofactor expansion.

Educational Mathematics

Educational mathematics involves presenting mathematical concepts in a way that is accessible and understandable for learners. Teaching methods are designed to build foundational skills and encourage problem-solving ability.

• Core concepts are broken into manageable steps.
• Visual aids and examples make abstract ideas concrete.
• Feedback and practice reinforce learning and build confidence.
• Connecting math to real-life examples helps engage students.

This approach finds use in teaching complex topics like matrix operations, enabling students to grasp and apply these concepts effectively.