Problem 26 - 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. \(\operatorname{det}\left(B^{5}\right)\)

Step-by-Step Solution

Answer

The determinant of \(B^5\) is \(1024\).

Step 1: Recall the Property for the Determinant of a Product

Recall the property that states: If \(A\) and \(B\) are square matrices of the same size, then \(\operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B)\).

Step 2: Apply the Property to Compute the Determinant of \(B^5\)

We need to compute \(\operatorname{det}\left(B^{5}\right)\). Since \(B^5\) is the product of five \(B\)'s, we can write \(\operatorname{det}\left(B^{5}\right)\) as: \(\operatorname{det}\left(B^{5}\right) = \operatorname{det}(B) \cdot \operatorname{det}(B) \cdot \operatorname{det}(B) \cdot \operatorname{det}(B) \cdot \operatorname{det}(B)\)

Step 3: Substitute the Given Value of \(\operatorname{det}(B)\) Into the Expression

We are given that \(\operatorname{det}(B) = -4\). Substitute this value into the expression found in Step 2: \(\operatorname{det}\left(B^{5}\right) = (-4) \cdot (-4) \cdot (-4) \cdot (-4) \cdot (-4)\)

Step 4: Simplify the Expression to Find the Determinant

Calculate the product in the expression to simplify it and obtain the final value for the determinant: \(\operatorname{det}\left(B^{5}\right) = 1024\) The determinant of \(B^5\) is \(1024\).

Key Concepts

Properties of Determinants

The properties of determinants help simplify complex calculations in linear algebra, making them vital tools in matrix computations. One fundamental property is that if you have two square matrices, say \(A\) and \(B\), the determinant of their product is the product of their determinants. This is expressed as:

• \( \operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B) \)

This property is extremely useful for handling higher powers of a matrix. For instance, if you want to find the determinant of \(B^n\) for any integer \(n\), each factor of \(B\) in the power contributes to the resulting determinant. Thus, the determinant of \(B^5\), which comprises five \(B\) matrices multiplied together, equals \((\operatorname{det}(B))^5\).
Understanding these properties makes it easier to tackle a range of matrix problems without manually multiplying the matrices first.

Matrix Multiplication

Matrix multiplication is a cornerstone of linear algebra that involves a sequence of operations where entries of corresponding rows and columns are multiplied and summed. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second.
Let's consider matrices \(A\) and \(B\):

• A is a matrix with dimensions \(m \times n\)
• B is a matrix with dimensions \(n \times p\)

Their product \(AB\) would then be an \(m \times p\) matrix. Each element of \(AB\) is derived by taking the dot product of corresponding rows of \(A\) and columns of \(B\).

• For entry \((i,j)\) of \(AB\), compute as \( a_{i1}b_{1j} + a_{i2}b_{2j} + \ldots + a_{in}b_{nj} \)

This process requires accurate matching of dimensions and maintaining order of multiplication, as matrix operations are not generally commutative; that is, \(AB eq BA\) in most cases.
Mastering matrix multiplication is essential for further understanding matrix transformations and solving linear equations.

3x3 Matrices

A 3x3 matrix is a square matrix that has three rows and three columns. It is often used in various applications such as rotations in 3D graphics, solving systems of equations, and more. Calculating determinants for a 3x3 matrix is slightly more complex than for a 2x2 matrix. Below is a helpful strategy to break it down:

• Take the elements of the first row, and each forms a mini-determinant with the corresponding two-by-two block that doesn't include that row and column.
• These determinants involve adding and subtracting the products of diagonals in their two-by-two configurations.
• Mathematically, you can express it for a matrix \(M = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \) as: \[ \operatorname{det}(M) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

These calculations reflect how dependencies within rows and columns of a matrix relate to its determinant. Efficiently managing these operations is key to handling more advanced problems in linear algebra.