Question

For the matrices $A=\left[\begin{array}{rr}2 & -3 \\ 4 & 1\end{array}\right]$ and $B=\left[\begin{array}{rr}0 & 5 \\ 1 & -2\end{array}\right]$ in $M_{2,2},$ determine whether the given matrix is a linear combination of $A$ and $B$. $\left[\begin{array}{rr}6 & 2 \\ 9 & 11\end{array}\right]$

          For the matrices
$A=\left[\begin{array}{rr}2 & -3 \\ 4 & 1\end{array}\right]$ and $B=\left[\begin{array}{rr}0 & 5 \\ 1 & -2\end{array}\right]$
in $M_{2,2},$ determine whether the given matrix is a linear combination of $A$ and $B$.

$\left[\begin{array}{rr}6 & 2 \\ 9 & 11\end{array}\right]$

Added by Patrick E.

Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J.… 5th Edition

Instant Answer

Solved by Expert Pritesh Ranjan

Step 1

$$ We can write this as a system of equations: \begin{align*} 2x + y &= 6, \\ -3x + 5y &= 2, \\ 4x + y &= 9, \\ x - 2y &= 11. \end{align*} We can solve this system using any method we like, but we'll use elimination. Adding the first and third equations, we get Show more…

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For the matrices $A=\left[\begin{array}{rr}2 & -3 \\ 4 & 1\end{array}\right]$ and $B=\left[\begin{array}{rr}0 & 5 \\ 1 & -2\end{array}\right]$ in $M_{2,2},$ determine whether the given matrix is a linear combination of $A$ and $B$. $\left[\begin{array}{rr}6 & 2 \\ 9 & 11\end{array}\right]$