Problem 2 - Intermediate Algebra

Question

For Problems \(1-10\), use expansion by minors to evaluate each determinant. (Objective 1) $$ \left|\begin{array}{rrr} 2 & 4 & 1 \\ -1 & 5 & 1 \\ -3 & 6 & 2 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 13.

Step 1: Identify the Matrix

The given matrix is a 3x3 matrix:\[\begin{pmatrix}2 & 4 & 1 \-1 & 5 & 1 \-3 & 6 & 2\end{pmatrix}\]

Step 2: Expansion by Minors - Selecting the First Row

We'll expand along the first row: \(2, 4, 1\).

Step 3: Calculate the Minor for Element (1,1)

For the element at position (1,1) (which is 2), remove the first row and first column and calculate the determinant of\[\left|\begin{array}{cc}5 & 1 \6 & 2\end{array}\right|= (5)(2) - (1)(6)= 10 - 6= 4.\]

Step 4: Determine the Cofactor for Element (1,1)

The cofactor is calculated by applying the sign rule: \((-1)^{1 + 1} \times 4 = 1 \times 4 = 4\).

Step 5: Calculate the Minor for Element (1,2)

For the element at position (1,2) (which is 4), remove the first row and second column and calculate the determinant of\[\left|\begin{array}{cc}-1 & 1 \-3 & 2\end{array}\right|= (-1)(2) - (1)(-3)= -2 + 3= 1.\]

Step 6: Determine the Cofactor for Element (1,2)

The cofactor is calculated by applying the sign rule: \((-1)^{1 + 2} \times 1 = -1 \times 1 = -1\).

Step 7: Calculate the Minor for Element (1,3)

For the element at position (1,3) (which is 1), remove the first row and third column and calculate the determinant of\[\left|\begin{array}{cc}-1 & 5 \-3 & 6\end{array}\right|= (-1)(6) - (5)(-3)= -6 + 15= 9.\]

Step 8: Determine the Cofactor for Element (1,3)

The cofactor is calculated by applying the sign rule: \((-1)^{1 + 3} \times 9 = 1 \times 9 = 9\).

Step 9: Combine the Cofactors to find the Determinant

Now, combine the results of the cofactors using the formula:\[\text{Determinant} = 2(4) + 4(-1) + 1(9)\]\[= 8 - 4 + 9= 13.\]

Key Concepts

Expansion by Minors

In mathematics, the method of expansion by minors is a process used to calculate the determinant of a matrix. This approach allows you to simplify the calculation by breaking it into smaller, more manageable steps.

• To start, you select a row or a column of the matrix. Any row or column can be used, but strategic selection may simplify calculations.
• In each step, you remove the chosen row and column that intersects with the element you're examining, which leaves you with a smaller sub-matrix.
• You then find the determinant of this sub-matrix, which is known as the minor of the element.

This is particularly useful for larger matrices where calculating the determinant directly is more complex. Expansion by minors reduces the problem into smaller parts that are easier to solve.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. It provides a lot of information about the matrix, such as whether it's invertible or not.

• To determine if a 3x3 matrix is invertible, simply check if the determinant is non-zero. If it isn't, the matrix doesn't have an inverse.
• The formula for expanding along the first row for a 3x3 matrix is: \[ \text{Det}(A) = a(\text{Minor of } a) + b(\text{Minor of } b) + c(\text{Minor of } c) \]

When calculated using expansion by minors, the determinant involves several steps of removing rows and columns and finding smaller determinants, as illustrated. This characteristic value plays a crucial role in fields such as linear algebra, where it's used in solving systems of equations, among other applications.

Cofactor Calculation

Calculating cofactors is an essential part of finding a determinant through expansion by minors. Each cofactor is associated with a minor, but it also includes a sign depending on the position of the element in the matrix.

• The sign of a cofactor is determined by the formula \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.
• For instance, the cofactor of an element in the first row and first column (1,1) is the minor multiplied by \((-1)^{1+1}\), which is 1.
• This sign pattern forms a checkerboard across the matrix, guiding which minors are subtracted and which are added when calculating the determinant.

By using these signed minors, we correctly compute the determinant through expansion by minors. This process ensures that all positional relationships within the matrix are preserved, leading to the accurate calculation of its determinant.