Given matrices P=[12b−a−35]*P*=[12−3​*b*−*a*5​] and Q=[4−4a55]*Q*=[45​−4*a*5​], if P=QT*P*=*QT*, find the value of 3a+2b3*a*+2*b*. beserta jawaban dan jalannya

We are given two matrices \( P \) and \( Q \), and the condition that \( P = Q^T \). The matrices are as follows:

\[
P = \begin{bmatrix} 1 & 2b \\ -a & -3 \end{bmatrix}, \quad Q = \begin{bmatrix} 4 & -4a \\ 5 & 5 \end{bmatrix}
\]

### Step 1: Write the transpose of \( Q \)
The transpose of \( Q \), denoted \( Q^T \), is obtained by swapping the rows and columns of \( Q \):

\[
Q^T = \begin{bmatrix} 4 & 5 \\ -4a & 5 \end{bmatrix}
\]

### Step 2: Set up the equation \( P = Q^T \)
From the problem, we are given that \( P = Q^T \), so we can set the elements of the two matrices equal to each other:

\[
\begin{bmatrix} 1 & 2b \\ -a & -3 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ -4a & 5 \end{bmatrix}
\]

### Step 3: Equate corresponding elements
By comparing the corresponding elements of the matrices, we obtain the following system of equations:

1. \( 1 = 4 \) (top-left element)
2. \( 2b = 5 \) (top-right element)
3. \( -a = -4a \) (bottom-left element)
4. \( -3 = 5 \) (bottom-right element)

### Step 4: Analyze the system of equations
From the first and fourth equations:

1. \( 1 = 4 \) is **impossible**, so there seems to be a typo or mistake in the problem setup. Let's assume it's a misprint and proceed with the other equations for now.

From the second equation:

\[
2b = 5 \implies b = \frac{5}{2}
\]

From the third equation:

\[
-a = -4a \implies a = 0
\]

### Step 5: Compute \( 3a + 2b \)
Now that we know \( a = 0 \) and \( b = \frac{5}{2} \), we can calculate:

\[
3a + 2b = 3(0) + 2\left(\frac{5}{2}\right) = 0 + 5 = 5
\]

### Final Answer:
The value of \( 3a + 2b \) is \( \boxed{5} \).

---

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. If \( P \) and \( Q \) were 3x3 matrices, how would you compute \( P = Q^T \)?
2. Can you find a condition where two matrices are symmetric?
3. What is the general formula for the transpose of a matrix?
4. How do you verify if two matrices are inverses of each other?
5. What are the properties of matrix multiplication?

**Tip:** When comparing matrices element-wise, always ensure that both matrices have the same dimensions.

Given matrices P=[12b−a−35] and Q=[4−4a55], if P=QT, find the value of 3a+2b.

This problem explores matrix transposition and solving for variables in the equation P = Q^T. Given two 2x2 matrices, P and Q, learn how to calculate the transpose, compare elements, and find the value of 3a + 2b. Suitable for students in Grades 10-12, the solution includes algebraic techniques and matrix properties.

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