Math  /  Algebra

Questionn the matrices AA and BB shown below, find 16A+B-\frac{1}{6} A+B. A=[2436]B=[82]A=\left[\begin{array}{l} -24 \\ -36 \end{array}\right] \quad B=\left[\begin{array}{l} -8 \\ -2 \end{array}\right]

Studdy Solution

STEP 1

What is this asking? We need to multiply matrix AA by 1/6-1/6, then add the result to matrix BB. Watch out! Make sure to multiply *all* entries of AA by 1/6-1/6, not just the first one!
Also, remember that we're *adding* BB to the result, not subtracting it.

STEP 2

1. Multiply AA by 1/6-1/6.
2. Add the result to BB.

STEP 3

We're **multiplying** each element of matrix AA by the **scalar** 1/6-1/6.
This is like distributing the 1/6-1/6 across the matrix.
This gives us a new matrix!
Let's call it CC.

STEP 4

c11=16(24)=246=4\begin{aligned} c_{11} &= -\frac{1}{6} \cdot (-24) \\ &= \frac{24}{6} \\ &= 4 \end{aligned} So, the **first element** of our new matrix CC is **4**!

STEP 5

c21=16(36)=366=6\begin{aligned} c_{21} &= -\frac{1}{6} \cdot (-36) \\ &= \frac{36}{6} \\ &= 6 \end{aligned} The **second element** of matrix CC is **6**!
Awesome!

STEP 6

So, our new matrix CC looks like this: C=[46]C = \begin{bmatrix} 4 \\ 6 \end{bmatrix}

STEP 7

Now, we're going to **add** matrix CC to matrix BB.
When adding matrices, we add corresponding elements.
Super simple!

STEP 8

We add the **first element** of CC to the **first element** of BB: 4+(8)=48=4\begin{aligned} 4 + (-8) &= 4 - 8 \\ &= -4 \end{aligned} So, the **first element** of our **final matrix** is **-4**!

STEP 9

We add the **second element** of CC to the **second element** of BB: 6+(2)=62=4\begin{aligned} 6 + (-2) &= 6 - 2 \\ &= 4 \end{aligned} The **second element** of our **final matrix** is **4**!
We're almost there!

STEP 10

Let's call our final matrix FF.
It looks like this: F=[44]F = \begin{bmatrix} -4 \\ 4 \end{bmatrix}

STEP 11

Our final answer, matrix FF, is: 16A+B=[44]-\frac{1}{6}A + B = \begin{bmatrix} -4 \\ 4 \end{bmatrix}

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