Consider the 3x3 matrices A and B, where det A = 2 and det B = 5. What is det [(3A)^-1B^2]?

A. 25/6
B. -675/2
C. 1350
D. 25/54

We are given two 3x3 matrices \( A \) and \( B \), with \( \det(A) = 2 \) and \( \det(B) = 5 \). We need to find \( \det \left( (3A)^{-1} B^2 \right) \).

### Step 1: Determinant properties
We will use the following properties of determinants:
1. \( \det(AB) = \det(A) \cdot \det(B) \) for any square matrices \( A \) and \( B \).
2. \( \det(A^{-1}) = \frac{1}{\det(A)} \) for an invertible matrix \( A \).
3. \( \det(cA) = c^n \det(A) \) for an \( n \times n \) matrix \( A \) and scalar \( c \), where \( n \) is the dimension of the matrix.

### Step 2: Determinant of \( (3A)^{-1} \)
First, calculate the determinant of \( 3A \):
\[
\det(3A) = 3^3 \det(A) = 27 \cdot 2 = 54
\]
Now, the determinant of \( (3A)^{-1} \) is:
\[
\det \left( (3A)^{-1} \right) = \frac{1}{\det(3A)} = \frac{1}{54}
\]

### Step 3: Determinant of \( B^2 \)
We know that \( \det(B^2) = (\det(B))^2 \). Therefore:
\[
\det(B^2) = 5^2 = 25
\]

### Step 4: Determinant of the product
Now, use the property \( \det(AB) = \det(A) \cdot \det(B) \):
\[
\det \left( (3A)^{-1} B^2 \right) = \det \left( (3A)^{-1} \right) \cdot \det(B^2) = \frac{1}{54} \cdot 25 = \frac{25}{54}
\]

Thus, the answer is:
\[
\boxed{D. \frac{25}{54}}
\]

Would you like a deeper explanation of any step? 

Here are some related questions to explore:
1. How do you find the inverse of a matrix?
2. What are the properties of matrix multiplication with determinants?
3. How does the determinant change under scalar multiplication?
4. How can determinants help in solving systems of linear equations?
5. What is the geometric interpretation of the determinant of a matrix?

**Tip:** Understanding matrix determinant properties simplifies many problems involving matrices.

Consider the 3x3 matrices A and B, where det A = 2 and det B = 5. What is det [(3A)^-1B^2]?

This solution explores the determinants of 3x3 matrices A and B, where det(A) = 2 and det(B) = 5. Learn how to compute det[(3A)^-1B^2] using key matrix properties such as scalar multiplication, inverse determinants, and the product of determinants. Suitable for students studying Linear Algebra at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation