Problem

Find the matrix product AB, if it is defined.
\[
A=\left[\begin{array}{rr}
0 & -2 \\
2 & 3
\end{array}\right], B=\left[\begin{array}{rr}
-2 & 0 \\
-1 & 1
\end{array}\right]
\]

Answer

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Answer

Final Answer: \(\boxed{\left[\begin{array}{rr} 2 & -2 \\ -7 & 3 \end{array}\right]}\)

Steps

Step 1 :We are given two matrices A and B. Matrix A is a 2x2 matrix with elements 0, -2, 2, 3. Matrix B is also a 2x2 matrix with elements -2, 0, -1, 1.

Step 2 :We are asked to find the matrix product AB. The matrix product AB is defined as the matrix C where each element c_ij is the dot product of the i-th row of A and the j-th column of B.

Step 3 :Let's calculate the elements of matrix C. The first element c_11 is the dot product of the first row of A and the first column of B. This gives us 2. The second element c_12 is the dot product of the first row of A and the second column of B. This gives us -2.

Step 4 :The third element c_21 is the dot product of the second row of A and the first column of B. This gives us -7. The fourth element c_22 is the dot product of the second row of A and the second column of B. This gives us 3.

Step 5 :So, the matrix product AB is a 2x2 matrix C with elements 2, -2, -7, 3.

Step 6 :Final Answer: \(\boxed{\left[\begin{array}{rr} 2 & -2 \\ -7 & 3 \end{array}\right]}\)

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