Problem 4 - College Algebra

Question

The determinant of \(2 \times 2\) matrix \(A\) is \(3 .\) If you switch the rows and multiply the fi st row by 6 and the second row by 2 , explain how to fi \(\mathrm{d}\) the determinant and provide the answer.

Step-by-Step Solution

Answer

The determinant is -36.

Step 1: Understand the Original Matrix

We have a \(2 \times 2\) matrix \(A\) with a given determinant of \(3\). The structure of the matrix \(A\) is generally: \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), and the determinant is calculated by the formula \(det(A) = ad - bc\).

Step 2: Swap the Rows

When you swap the rows of a \(2 \times 2\) matrix, the determinant of the new matrix is the negative of the original determinant. If the original determinant is \(3\), after swapping rows, the determinant becomes \(-3\).

Step 3: Multiply the First Row by 6

Multiplying elements of a row in a matrix by a scalar multiplies the determinant by that same scalar. Multiply the first row by \(6\) changes the determinant to \(-3 \times 6 = -18\).

Step 4: Multiply the Second Row by 2

Similarly, multiplying the elements of the second row by \(2\) further multiplies the determinant by the same scalar. Therefore, the new determinant becomes \(-18 \times 2 = -36\).

Step 5: Finalize the Determinant

After swapping rows and applying the row multiplications, calculate the resulting determinant as \(-36\). This is the determinant of the transformed matrix.

Key Concepts

Matrix Operations

Working with matrices involves several operations that help in transforming and solving matrix-related problems. In a typical matrix, like a \(2 \times 2\) matrix, operations such as swapping rows, multiplying rows by scalars, and others, can significantly alter the characteristics of the matrix.
When engaging in matrix operations, you often deal with the determinant, which is a special number calculated from its elements. The determinant gives insights into the matrix characteristics, like whether it's invertible. For a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is computed as \(ad - bc\).
Understanding these operations is crucial as they help in transforming matrices for various applications in mathematics, science, and engineering. Each operation affects the matrix in different ways; hence, correctly applying them is essential for accurate results.

Row Multiplication in Matrices

Row multiplication is a key matrix operation that involves multiplying all the elements in a row by a nonzero scalar. This operation affects the determinant of the matrix. Suppose you have a \(2 \times 2\) matrix, and you multiply the first row by 6. This transforms the determinant by multiplying it with the scalar.
For example, if the original determinant of your matrix is 3, after multiplying the first row by 6, the determinant becomes \(3 \times 6 = 18\). This principle is applicable to any row in a matrix; multiplying any row by a scalar will result in the determinant being multiplied by that scalar.

• The determinant indicates matrix properties like area in geometric interpretations, so altering it via row operations changes these properties accordingly.
• Double-check these operations to avoid computational errors, as they can greatly influence the final outcome.

Row multiplication is simple yet powerful, providing flexibility in matrix transformations.

Swapping Rows Effect on Determinant

Swapping rows in a matrix is another fundamental operation that changes the sign of the determinant. This operation can be crucial when you’re simplifying or transforming a matrix for further calculations.
Consider, for instance, a \(2 \times 2\) matrix with a determinant of 3. By swapping its rows, you instantly change the determinant sign, making it \(-3\). This effect occurs universally; swapping any two rows in a matrix results in a sign change of the determinant without altering its absolute value.

• This operation doesn’t affect the magnitude, only the sign, hence it’s useful in solving systems where determinant direction is important.
• Ensure that after swapping, further determinant changes (like row multiplication) happen correctly based on the new row arrangement.

Understanding this concept makes matrix handling easier, especially in solving linear algebra problems.