Math  /  Algebra

QuestionGiven the matrices AA and BB shown below, find 12B+4A\frac{1}{2} B+4 A. A=[332414]B=[10141014422]A=\left[\begin{array}{cc} 3 & -3 \\ 2 & 4 \\ 1 & 4 \end{array}\right] \quad B=\left[\begin{array}{cc} 10 & -14 \\ -10 & -14 \\ -4 & 22 \end{array}\right]

Studdy Solution

STEP 1

What is this asking? We're taking two matrices, AA and BB, multiplying them by some numbers, and then adding them together! Watch out! Remember, we can only add matrices of the same dimensions, and these are both 3x2, so we’re good to go!

STEP 2

1. Multiply Matrix BB by 12\frac{1}{2}
2. Multiply Matrix AA by 44
3. Add the Results

STEP 3

Let's **multiply** each element of matrix BB by 12\frac{1}{2}.
Why? Because that's what it means to multiply a matrix by a scalar!
This **scales** each part of the matrix.

STEP 4

12B=12[10141014422]=[5757211] \frac{1}{2} B = \frac{1}{2} \left[\begin{array}{cc} 10 & -14 \\ -10 & -14 \\ -4 & 22 \end{array}\right] = \left[\begin{array}{cc} 5 & -7 \\ -5 & -7 \\ -2 & 11 \end{array}\right]

STEP 5

Now, we'll **multiply** each element of matrix AA by 44.
Remember, multiplying a matrix by a scalar means multiplying every single element by that scalar.

STEP 6

4A=4[332414]=[1212816416] 4A = 4 \left[\begin{array}{cc} 3 & -3 \\ 2 & 4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{cc} 12 & -12 \\ 8 & 16 \\ 4 & 16 \end{array}\right]

STEP 7

Finally, we **add** the two resulting matrices together.
We do this element by element – adding the numbers in the same position in each matrix.

STEP 8

12B+4A=[171939227] \frac{1}{2}B + 4A = \left[\begin{array}{cc} 17 & -19 \\ 3 & 9 \\ 2 & 27 \end{array}\right]

STEP 9

So, 12B+4A=[171939227]\frac{1}{2}B + 4A = \left[\begin{array}{cc} 17 & -19 \\ 3 & 9 \\ 2 & 27 \end{array}\right].
We did it!

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