To prove that \( a = b \) given the matrix equation \(\left[\begin{array}{ll} a & b \end{array}\right]\left[\begin{array}{l} 2 \\ 3 \end{array}\right] = \left[\begin{array}{ll} 1 & 4 \end{array}\right]\left[\begin{array}{l} a \\ b \end{array}\right]\):
Step-by-step, we follow these steps:
1. Compute the left-hand side of the equation:
[tex]\[
\left[\begin{array}{ll} a & b \end{array}\right]\left[\begin{array}{l} 2 \\ 3 \end{array}\right]
\][/tex]
This is done by performing the dot product:
[tex]\[
a \cdot 2 + b \cdot 3 = 2a + 3b
\][/tex]
2. Compute the right-hand side of the equation:
[tex]\[
\left[\begin{array}{ll} 1 & 4 \end{array}\right]\left[\begin{array}{l} a \\ b \end{array}\right]
\][/tex]
This is also done by performing the dot product:
[tex]\[
1 \cdot a + 4 \cdot b = a + 4b
\][/tex]
3. Set the two expressions equal to each other:
[tex]\[
2a + 3b = a + 4b
\][/tex]
4. Solve for \( a \) and \( b \):
Let's isolate \( a \). Start by subtracting \( a \) and \( 3b \) from both sides:
[tex]\[
2a + 3b - a - 3b = a + 4b - a - 3b
\][/tex]
Simplifying:
[tex]\[
a = b
\][/tex]
Thus, we have shown that [tex]\( a = b \)[/tex] based on the given matrix equation.
