Math  /  Algebra

Question1\checkmark 1 2 3 4 5\checkmark 5 6
Let A=[1223]A=\left[\begin{array}{cc}1 & 2 \\ -2 & -3\end{array}\right] and B=[3011]B=\left[\begin{array}{cc}3 & 0 \\ -1 & -1\end{array}\right]. Find each matrix below. If a matrix is not defined, click on "Undefined".

Studdy Solution

STEP 1

What is this asking? We're given two matrices, A A and B B , and we need to find their product, AB A \cdot B . Watch out! Be careful with matrix multiplication!
It's not like regular multiplication.
The order matters, and you have to make sure the dimensions match up!

STEP 2

1. Check dimensions
2. Multiply matrices

STEP 3

First, let's **check the dimensions** of matrices A A and B B .
Matrix A A is a 2×2 2 \times 2 matrix, and matrix B B is also a 2×2 2 \times 2 matrix.
Since the number of columns in A A matches the number of rows in B B , we can multiply them!

STEP 4

Now, let's **multiply the matrices**!
We'll use the formula for matrix multiplication: the element in the i i -th row and j j -th column of the product matrix is the dot product of the i i -th row of the first matrix and the j j -th column of the second matrix.

STEP 5

Let's calculate the first element of the product matrix, which is in the first row and first column:
(13)+(21)=32=1(1 \cdot 3) + (2 \cdot -1) = 3 - 2 = 1

STEP 6

Next, the element in the first row and second column:
(10)+(21)=02=2(1 \cdot 0) + (2 \cdot -1) = 0 - 2 = -2

STEP 7

Now, the element in the second row and first column:
(23)+(31)=6+3=3(-2 \cdot 3) + (-3 \cdot -1) = -6 + 3 = -3

STEP 8

Finally, the element in the second row and second column:
(20)+(31)=0+3=3(-2 \cdot 0) + (-3 \cdot -1) = 0 + 3 = 3

STEP 9

The product matrix AB A \cdot B is:
[1233]\left[\begin{array}{cc}1 & -2 \\ -3 & 3\end{array}\right]

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