Paul Bolton
04/09/2024 · Junior High School
Determine the value of \( a, b, c \) and \( d \) in the following matrix equation: \[ \left[\begin{array}{cc}2 & 3 \\ -1 & -1\end{array}\right]+a\left[\begin{array}{ll}2 & 1 \\ 0 & 4\end{array}\right]=\left[\begin{array}{ll}8 & b \\ c & d\end{array}\right] \]
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve for \( a, b, c, \) and \( d \) in the matrix equation
\[
\left[\begin{array}{cc}2 & 3 \\ -1 & -1\end{array}\right]+a\left[\begin{array}{ll}2 & 1 \\ 0 & 4\end{array}\right]=\left[\begin{array}{ll}8 & b \\ c & d\end{array}\right],
\]
we start by performing the matrix addition on the left-hand side.
First, we calculate \( a \left[\begin{array}{ll}2 & 1 \\ 0 & 4\end{array}\right] \):
\[
a \left[\begin{array}{ll}2 & 1 \\ 0 & 4\end{array}\right] = \left[\begin{array}{ll}2a & a \\ 0 & 4a\end{array}\right].
\]
Now, we add this result to the first matrix:
\[
\left[\begin{array}{cc}2 & 3 \\ -1 & -1\end{array}\right] + \left[\begin{array}{ll}2a & a \\ 0 & 4a\end{array}\right] = \left[\begin{array}{cc}2 + 2a & 3 + a \\ -1 + 0 & -1 + 4a\end{array}\right].
\]
This simplifies to:
\[
\left[\begin{array}{cc}2 + 2a & 3 + a \\ -1 & -1 + 4a\end{array}\right].
\]
Now we set this equal to the right-hand side of the original equation:
\[
\left[\begin{array}{cc}2 + 2a & 3 + a \\ -1 & -1 + 4a\end{array}\right] = \left[\begin{array}{ll}8 & b \\ c & d\end{array}\right].
\]
From this equality, we can equate the corresponding elements:
1. From the first element:
\[
2 + 2a = 8.
\]
Solving for \( a \):
\[
2a = 8 - 2 \implies 2a = 6 \implies a = 3.
\]
2. From the second element:
\[
3 + a = b.
\]
Substituting \( a = 3 \):
\[
3 + 3 = b \implies b = 6.
\]
3. From the third element:
\[
-1 = c.
\]
Thus, we have:
\[
c = -1.
\]
4. From the fourth element:
\[
-1 + 4a = d.
\]
Substituting \( a = 3 \):
\[
-1 + 4 \cdot 3 = d \implies -1 + 12 = d \implies d = 11.
\]
Now we have determined the values of \( a, b, c, \) and \( d \):
\[
\boxed{3}, \quad \boxed{6}, \quad \boxed{-1}, \quad \boxed{11}.
\]
Quick Answer
The values of \( a, b, c, \) and \( d \) are \( a=3, b=6, c=-1, d=11 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit