Problem 42 - College Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{rrrr}3 & -1 & 1 & 2 \\\\-2 & 0 & 0 & 0 \\\2 & -1 & -2 & 3 \\\1 & 4 & 2 & 3\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -48.

Step 1: Initial Matrix

Start from the original matrix: \[ \begin{vmatrix} 3 & -1 & 1 & 2 \ -2 & 0 & 0 & 0 \ 2 & -1 & -2 & 3 \ 1 & 4 & 2 & 3 \ \end{vmatrix} \]

Step 2: Row addition operation

Perform the elementary row operation R2 = R2+2*R1, which makes the element in the (2,1) position zero. \[ \begin{vmatrix} 3 & -1 & 1 & 2 \ 0 & -2 & 2 & 4 \ 2 & -1 & -2 & 3 \ 1 & 4 & 2 & 3 \ \end{vmatrix} \]

Step 3: Further row additions

Further row operations R3 = R3-2*R1 and R4 = R4-R1 to zero the elements in the (3,1) and (4,1) positions respectively. \[ \begin{vmatrix} 3 & -1 & 1 & 2 \ 0 & -2 & 2 & 4 \ 0 & 1 & -4 & -1 \ 0 & 5 & 1 & 1 \ \end{vmatrix} \]

Step 4: Finally lower triangular form

Using R4 = R4-5*R2 to make element in the (4,2) position zero, we get lower triangular form \[ \begin{vmatrix} 3 & -1 & 1 & 2 \ 0 & -2 & 2 & 4 \ 0 & 0 & -4 & -1 \ 0 & 0 & -2 & -2\ \end{vmatrix} \]

Step 5: Evaluate determinant

The determinant of a lower triangular matrix is the product of the elements on the main diagonal. Hence determinant = 3 * -2 * -4 * -2 = -48.

Key Concepts

Elementary Row Operations

Elementary row operations are crucial tools in matrix algebra. They help us modify matrices without changing their fundamental properties, such as determinants. Elementary row operations include:

• Row swapping Moving entire rows around.
• Row multiplication Multiplying a row by a non-zero constant.
• Row addition Adding or subtracting a multiple of one row from another.

These operations can simplify matrices, making calculations more manageable. For instance, in the original problem, we used row addition to create zeros below the pivot (the leading non-zero entry in a row). Achieving zeros, especially in a column, is crucial in simplifying to a form like a triangular matrix, which allows for easier determinant calculations.

Matrix Algebra

Matrix algebra is a field of mathematics that explores the operations and characteristics of matrices. It includes various methods for solving linear equations, finding determinants, and dealing with matrix transformations.

Key operations in matrix algebra include:

• Addition and subtraction of matrices Performed element-wise when matrices are of the same dimension.
• Matrix multiplication Combining two matrices to produce a third matrix following row-by-column rules.
• Finding determinants A scalar value that summarizes certain properties of a matrix, like invertibility.

Matrix algebra is foundational in understanding linear transformations and systems of equations, making it essential for calculus and advanced algebra applications.

Triangular Matrix

A triangular matrix is a type of square matrix where all entries above or below the main diagonal are zero. If the elements below the diagonal are zero, it's called an upper triangular matrix. Conversely, if elements above the diagonal are zero, it's a lower triangular matrix.

Triangular matrices simplify various matrix operations. Notably, the determinant of a triangular matrix is simply the product of its diagonal elements. This makes evaluating determinants easier compared to a generic matrix.

In our exercise, converting a matrix into a lower triangular form by using elementary row operations allowed for a straightforward computation of the determinant. We found out that the determinant was simply the product of the diagonal elements, which is quicker than calculating it directly from the original matrix.