https://file.aiquickdraw.com/mathbot/file/17333701344014js8yqpg.jpg

The problem is asking you to work with matrices \( A = \begin{pmatrix} 3 & -4 \\ 1 & 2 \end{pmatrix} \) and \( B = \begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix} \), to compute the following:

1. \( B^T \) (the transpose of \( B \))
2. \( A^{-1} \) (the inverse of \( A \))
3. \( A^T + 3B \) (the transpose of \( A \) added to 3 times \( B \))

### Solutions:

#### 1. \( B^T \)
The transpose of a matrix is obtained by swapping rows and columns. Therefore:
\[
B = \begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix}, \quad B^T = \begin{pmatrix} 3 & 1 \\ 2 & 0 \end{pmatrix}.
\]

#### 2. \( A^{-1} \)
The inverse of \( A \) is computed as:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \text{ where } \text{det}(A) = ad - bc.
\]
For \( A = \begin{pmatrix} 3 & -4 \\ 1 & 2 \end{pmatrix} \):
\[
\text{det}(A) = (3)(2) - (1)(-4) = 6 + 4 = 10.
\]
So:
\[
A^{-1} = \frac{1}{10} \begin{pmatrix} 2 & 4 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 0.2 & 0.4 \\ -0.1 & 0.3 \end{pmatrix}.
\]

#### 3. \( A^T + 3B \)
The transpose of \( A \), \( A^T \), is:
\[
A^T = \begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix}.
\]
Now, calculate \( 3B \):
\[
3B = 3 \cdot \begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 9 & 6 \\ 3 & 0 \end{pmatrix}.
\]
Finally, compute \( A^T + 3B \):
\[
A^T + 3B = \begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix} + \begin{pmatrix} 9 & 6 \\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 12 & 7 \\ -1 & 2 \end{pmatrix}.
\]

### Final Results:
1. \( B^T = \begin{pmatrix} 3 & 1 \\ 2 & 0 \end{pmatrix} \)
2. \( A^{-1} = \begin{pmatrix} 0.2 & 0.4 \\ -0.1 & 0.3 \end{pmatrix} \)
3. \( A^T + 3B = \begin{pmatrix} 12 & 7 \\ -1 & 2 \end{pmatrix} \)

Would you like a detailed explanation of any of these steps? 

### Expanding Your Understanding:
1. How do you calculate the determinant of a matrix?
2. What is the geometric meaning of a transpose of a matrix?
3. How does the inverse of a matrix relate to solving linear equations?
4. Why does the determinant of a matrix need to be nonzero for the inverse to exist?
5. How does scalar multiplication affect each element of a matrix?

**Tip**: When finding the inverse of a matrix, always double-check the determinant value—it ensures the calculation is valid!

Diberi matriks A = [[3, -4], [1, 2]] dan B = [[3, 2], [1, 0]]. Cari nilai bagi: (i) B^T, (ii) A^-1, (iii) A^T + 3B.

Explore how to calculate the transpose of a matrix (B^T), the inverse of a matrix (A^-1), and perform matrix operations like A^T + 3B using the matrices A = [[3, -4], [1, 2]] and B = [[3, 2], [1, 0]]. This step-by-step guide explains essential matrix concepts, including the determinant and scalar multiplication.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation