VIPSolutionsAct like a helpful tutor and exlain me :Example 1: Solve the equation D = 2AB - C. 2.5 41 08 aefaile=[aa] ee]
Question:
Act like a helpful tutor and exlain me :
Example 1: Solve the equation D = 2AB - C.
2.5 41 08
aefaile=[aa] ee]
This is AI generated Response.
Asked by: Nithin Vanga
Created at: 2025-10-15 09:59:13
Asked by: Nithin Vanga
Created at: 2025-10-15 09:59:13
Final Answer:
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Certainly! Let's solve the equation step by step:
## Step 1: Write Down the Equation
Given:
\[
D = 2AB - C
\]
where
\( A = \begin{bmatrix} 2 & 5 \\ 9 & 1 \end{bmatrix} \)
\( B = \begin{bmatrix} 4 & 1 \\ 3 & 7 \end{bmatrix} \)
\( C = \begin{bmatrix} & 8 \\ 2 & 6 \end{bmatrix} \)
---
## Step 2: Find \( AB \)
Multiply matrices \( A \) and \( B \):
\[
AB =
\begin{matrix}
2 & 5 \\
9 & 1
\end{bmatrix}
\begin{bmatrix}
4 & 1 \\
3 & 7
\end{bmatrix}
\]
Multiply rows of \( A \) by columns of \( B \):
- First row, first column: \( 2 \times 4 + 5 \times 3 = 8 + 15 = 23 \)
- First row, second column: \( 2 \times 1 + 5 \times 7 = 2 + 35 = 37 \)
- Second row, first column: \( 9 \times 4 + 1 \times 3 = 36 + 3 = 39 \)
- Second row, second column: \( 9 \times 1 + 1 \times 7 = 9 + 7 = 16 \)
So,
\[
AB = \begin{bmatrix}
23 & 37 \\
39 & 16
\end{bmatrix}
\]
---
## Step 3: Multiply by 2
\[
2AB = 2 \times \begin{bmatrix}
23 & 37 \\
39 & 16
\end{bmatrix}
=
\begin{bmatrix}
46 & 74 \\
78 & 32
\end{bmatrix}
\]
---
## Step 4: Subtract Matrix \( C \)
\[
D = 2AB - C =
\begin{bmatrix}
46 & 74 \\
78 & 32
\end{bmatrix}
-
\begin{bmatrix}
& 8 \\
2 & 6
\end{bmatrix}
=
\begin{bmatrix}
46- & 74-8 \\
78-2 & 32-6
\end{bmatrix}
=
\begin{bmatrix}
46 & 66 \\
76 & 26
\end{bmatrix}
\]
---
## **Final Answer**
\[
\boxed{
D = \begin{bmatrix}
46 & 66 \\
76 & 26
\end{bmatrix}
}
\]
Let me know if you need more explanation on any step!
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