matriks a=(-3 0 1 5) b=(4 -1 3 1), C=AB, matriks invers c adalah

Given the matrices:

- \( A = \begin{pmatrix} -3 & 0 \\ 1 & 5 \end{pmatrix} \)
- \( B = \begin{pmatrix} 4 & -1 \\ 3 & 1 \end{pmatrix} \)

The matrix \( C = A \times B \) is calculated by performing matrix multiplication of \( A \) and \( B \).

### Step 1: Matrix Multiplication
To compute \( C = AB \), we use the rule for matrix multiplication, where each element of the resulting matrix is obtained by summing the products of corresponding entries from rows of \( A \) and columns of \( B \).

\[
C_{11} = (-3 \times 4) + (0 \times 3) = -12
\]
\[
C_{12} = (-3 \times -1) + (0 \times 1) = 3
\]
\[
C_{21} = (1 \times 4) + (5 \times 3) = 4 + 15 = 19
\]
\[
C_{22} = (1 \times -1) + (5 \times 1) = -1 + 5 = 4
\]

Thus, matrix \( C \) is:

\[
C = \begin{pmatrix} -12 & 3 \\ 19 & 4 \end{pmatrix}
\]

### Step 2: Inverse of Matrix \( C \)
To find the inverse of \( C \), we need to compute the determinant of \( C \) and then apply the formula for the inverse of a \( 2 \times 2 \) matrix.

The formula for the inverse of a \( 2 \times 2 \) matrix \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is:

\[
M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For matrix \( C \):

\[
\text{det}(C) = (-12 \times 4) - (3 \times 19) = -48 - 57 = -105
\]

Since the determinant is non-zero, the inverse of \( C \) exists. The inverse is:

\[
C^{-1} = \frac{1}{-105} \begin{pmatrix} 4 & -3 \\ -19 & -12 \end{pmatrix}
\]

Thus, the inverse of matrix \( C \) is:

\[
C^{-1} = \begin{pmatrix} \frac{4}{-105} & \frac{-3}{-105} \\ \frac{-19}{-105} & \frac{-12}{-105} \end{pmatrix}
= \begin{pmatrix} -\frac{4}{105} & \frac{1}{35} \\ \frac{19}{105} & \frac{4}{35} \end{pmatrix}
\]

Would you like more details, or have any questions?

Here are 5 related questions:
1. How do you find the determinant of a \( 3 \times 3 \) matrix?
2. What conditions must hold for a matrix to have an inverse?
3. How is matrix multiplication defined for \( 3 \times 3 \) matrices?
4. What is the geometric interpretation of matrix inverses?
5. How are matrix operations used in solving systems of equations?

**Tip**: The inverse of a matrix exists only if its determinant is non-zero.

Solve a problem involving matrix multiplication and matrix inversion. Learn how to compute the product of matrices A and B, and find the inverse of the resulting matrix C. This solution covers matrix multiplication rules, determinant calculation, and the inversion process for 2x2 matrices. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation