https://file.aiquickdraw.com/mathbot/file/1727321043321vpxkbfx8.jpg

We have the following matrix equation based on the image you provided:

\[
2 \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ c & 4 \end{pmatrix} \begin{pmatrix} 2 & d \\ 2 & 3 \end{pmatrix}
\]

We will first simplify the left-hand side and then solve the matrix multiplication on the right-hand side to compare and find the values of \(a\), \(b\), \(c\), and \(d\).

### Left-hand side simplification:

First, multiply the matrix by 2:

\[
2 \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} 2a & 4 \\ -6 & 2 \end{pmatrix}
\]

Now, add the matrix \(\begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix}\):

\[
\begin{pmatrix} 2a & 4 \\ -6 & 2 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 2a + 4 & 4 - 1 \\ -6 + 0 & 2 + b \end{pmatrix}
\]

This results in:

\[
\begin{pmatrix} 2a + 4 & 3 \\ -6 & 2 + b \end{pmatrix}
\]

### Right-hand side simplification:

Now, we compute the matrix product on the right-hand side:

\[
\begin{pmatrix} 3 & 2 \\ c & 4 \end{pmatrix} \begin{pmatrix} 2 & d \\ 2 & 3 \end{pmatrix}
\]

Performing matrix multiplication:

\[
\begin{pmatrix} (3 \times 2 + 2 \times 2) & (3 \times d + 2 \times 3) \\ (c \times 2 + 4 \times 2) & (c \times d + 4 \times 3) \end{pmatrix}
\]

This gives:

\[
\begin{pmatrix} 6 + 4 & 3d + 6 \\ 2c + 8 & cd + 12 \end{pmatrix} = \begin{pmatrix} 10 & 3d + 6 \\ 2c + 8 & cd + 12 \end{pmatrix}
\]

### Equating both sides:

We now compare the left-hand and right-hand sides:

\[
\begin{pmatrix} 2a + 4 & 3 \\ -6 & 2 + b \end{pmatrix} = \begin{pmatrix} 10 & 3d + 6 \\ 2c + 8 & cd + 12 \end{pmatrix}
\]

From this, we have the following system of equations:

1. \(2a + 4 = 10\)
2. \(3 = 3d + 6\)
3. \(-6 = 2c + 8\)
4. \(2 + b = cd + 12\)

#### Solving the system:

1. From \(2a + 4 = 10\), we solve for \(a\):
   \[
   2a = 6 \implies a = 3
   \]

2. From \(3 = 3d + 6\), we solve for \(d\):
   \[
   3d = -3 \implies d = -1
   \]

3. From \(-6 = 2c + 8\), we solve for \(c\):
   \[
   2c = -14 \implies c = -7
   \]

4. Substituting \(d = -1\) into the equation \(2 + b = cd + 12\):
   \[
   2 + b = (-7)(-1) + 12 \implies 2 + b = 7 + 12 = 19
   \]
   So,
   \[
   b = 17
   \]

### Sum of \(a + b + c + d\):

\[
a + b + c + d = 3 + 17 - 7 - 1 = 12
\]

It appears the sum is 12, but none of the options match this exactly. Could you clarify if there is an adjustment needed, or if there may be an issue with the provided question?

Do you need further details or clarifications?

Here are five relative questions:
1. How can you verify matrix multiplication results step-by-step?
2. What happens if there are multiple matrix solutions?
3. How do determinant and inverse matrices play a role in such equations?
4. What is the significance of non-commutative property in matrix multiplication?
5. How do we check if a matrix equation has no solution?

**Tip:** Always check matrix dimensions when performing multiplication to ensure compatibility.

Jika 2(a 2 -3 1) + (4 -1 0 b) = (3 2 c 4)(2 d 2 3), nilai a + b + c + d = ...

In this problem, we solve a matrix equation by performing matrix multiplication and addition. The problem involves finding the values of variables a, b, c, and d by equating both sides of the matrix equation and solving a system of linear equations. 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