Problem 12 - Differential Equations and Linear Algebra

Question

Suppose that \(A\) and \(B\) are \(4 \times 4\) invertible matrices. If \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(B)=3,\) compute each determinant below. $$\operatorname{det}(A B)$$.

Step-by-Step Solution

Answer

Based on the given information and properties of determinants, we can compute the determinant of the product of matrices A and B, denoted as det(AB), as follows: det(AB) = det(A) * det(B) = (-2) * (3) = \(-6\).

Step 1: Recall the property for determinants of matrix products

We should recall the following property for determinants: det(AB) = det(A) * det(B). This is applicable for any two square matrices A and B.

Step 2: Use given determinant values to compute det(AB)

We are given det(A) = -2 and det(B) = 3. Using the property from Step 1, we can compute the determinant of the product AB as: det(AB) = det(A) * det(B) = (-2) * (3)

Step 3: Calculate the value of det(AB)

Computing the product: det(AB) = (-2) * (3) = -6 The determinant of the product of matrices A and B, det(AB), is -6.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that involves combining two matrices to produce a new matrix. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. In this process, each element of the resulting matrix is calculated by taking the dot product of rows from the first matrix and columns from the second matrix.

When working with square matrices, such as the invertible matrices mentioned in the exercise, matrix multiplication adheres to the associative and distributive properties. It's crucial to understand that matrix multiplication is not commutative. This means that the order in which you multiply matrices matters -

• If you have two matrices, A and B, then generally, \(AB eq BA\) unless the matrices have specific properties that make them equal.
• When multiplied, matrices produce another matrix that stores the combined information of the original matrices.

Knowing how matrix multiplication interacts with determinants is essential for deeper algebraic operations.

Properties of Determinants

Determinants are an indispensable tool in linear algebra, providing important scalar value that reflect certain properties of matrices. They give insights into matrix characteristics such as invertibility and linear independence.

For any two square matrices, A and B, one significant property is that the determinant of their product is the product of their determinants:

• If you have determinants \(\text{det}(A)\) and \(\text{det}(B)\), then \(\text{det}(AB) = \text{det}(A) \times \text{det}(B)\).
• Determinants also possess properties like being zero when a matrix is non-invertible (which means it doesn't have a multiplicative inverse).
• Other properties include that multiplying a matrix by a scalar combusts the determinant by that scalar.

This property is particularly useful in exercises where determinant values of matrices and their products need to be computed quickly, as seen in the example.

Invertible Matrices

An invertible matrix, also known as a non-singular matrix, is a square matrix that possesses an inverse. The existence of an inverse is crucial for solving systems of linear equations, among other applications.

The determinant offers a straightforward test for invertibility:

• A matrix is invertible if and only if its determinant is not zero.
• In the context of the exercise, both matrices A and B are stated to be invertible.
• This automatically tells us \(\text{det}(A) eq 0\) and \(\text{det}(B) eq 0\), aligning with the given determinant values \(\text{det}(A) = -2\) and \(\text{det}(B) = 3\).

Recognizing and working with invertible matrices and their determinants aids in understanding broader concepts in linear algebra, such as linear transformations and eigenvalues.