Problem 28 - Differential Equations and Linear Algebra

Question

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}(C),\) where \(C\) is obtained from matrix \(B\) by interchanging the last two columns and multiplying the first column by 4

Step-by-Step Solution

Answer

The determinant of matrix C is 16.

Step 1: Determine matrix C from matrix B

First, we need to obtain matrix C from matrix B by: - Interchanging the second and third columns. - Multiplying the first column by 4. Let M be a matrix after interchanging the last two columns of B, and C be obtained from M by multiplying the first column by 4.

Step 2: Find det(M) using the property of exchanging columns

Since we have not yet multiplied any columns, det(M) just involves exchanging the last two columns of B. According to the property mentioned in the analysis, exchanging columns changes the sign of the determinant: \( \operatorname{det}(M) = -\operatorname{det}(B) \) Given that det(B) = -4, we have: \( \operatorname{det}(M) = -(-4) = 4 \)

Step 3: Find det(C) using the property of scalar multiplication

Now that we have det(M), we can find det(C) by considering the scalar multiplication. Since we multiply the first column of M by 4 to obtain the matrix C, we will also multiply the determinant of M by 4. So we have: \( \operatorname{det}(C) = 4 \times \operatorname{det}(M) \) Using the value of det(M) = 4, we get: \( \operatorname{det}(C) = 4 \times 4 \)

Step 4: Calculate det(C)

Now, we can simply calculate det(C) as: \( \operatorname{det}(C) = 4 \times 4 = 16 \) So, the determinant of matrix C is 16.

Key Concepts

Matrix Operations

Matrix operations include a variety of tasks such as addition, subtraction, multiplication, and transformations applied to matrices. When handling these operations, the dimensions of matrices must be compatible. For example, matrix multiplication is only possible if the number of columns in the first matrix matches the number of rows in the second.

Operations can also involve scalar multiplication and column or row transpositions. In scalar multiplication, each element of a matrix is multiplied by a constant, affecting properties like the determinant. Similarly, swapping columns or rows is a simple transformation that can change the sign of a determinant.

These operations are fundamental in many mathematical computations, illustrating how matrices can model and solve complex problems in engineering, physics, and computer science.

Determinant Properties

The determinant of a matrix is a special number that provides vital information about the matrix, such as whether it is invertible. For square matrices, determinants can be calculated using various methods, like Laplace expansion or leveraging the properties of triangular matrices.

Key properties of determinants include:

• Determinant sign change upon swapping rows or columns.
• Multiplying a row or column by a scalar results in the determinant being multiplied by the same scalar.
• If two rows or columns are identical, the determinant is zero.

Understanding these properties allows mathematicians to manipulate matrices effectively during transformations or complex operations.

Matrix Transformations

Matrix transformations encompass various operations that change a matrix while utilizing its inherent structure. These transformations can be simple, like switching columns or multiplying them by a constant, or more complex, involving row reduction techniques and eigenvalue calculations.

Transformations are crucial in solving systems of linear equations, performing computer graphics operations, and understanding the behavior of linear maps. A typical transformation discussed in the provided exercise involves interchanging columns and scaling a column by a constant factor, which directly influences the determinant of a matrix.

Learning how these transformations work and affect the matrix attributes is critical to mastering the determinant calculation and other advanced linear algebra topics.