Find answers to your questions and expand your knowledge with IDNLearn.com. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
If [tex]\[
\left[\begin{array}{ll} a & b \end{array}\right] \left[\begin{array}{l} 2 \\ 3 \end{array}\right] = \left[\begin{array}{ll} 1 & 4 \end{array}\right] \left[\begin{array}{l} a \\ b \end{array}\right],
\][/tex] then prove that [tex]\( a = b \)[/tex].
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.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your go-to source for accurate answers. Thanks for stopping by, and come back for more helpful information.
