Multiply the Matrices

Problem

[[3,1,−1],[0,−1,−1],[0,−1,−2],[0,2,1]]*[[3,1,−1],[0,−1,−1],[0,0,−1],[0,0,−1]]

Solution

1. Identify the dimensions of the matrices. The first matrix is 4×3 and the second matrix is 3×4 However, looking at the provided input, the second matrix is written as a 4×3 matrix as well. Since the number of columns in the first (3 must match the number of rows in the second, the second matrix must be interpreted as a 3×4 matrix:

B=[[3,1,−1,0],[0,−1,−1,0],[0,0,−1,−1]]

Wait, re-evaluating the input strings: the second matrix is provided as [[3,1,-1],[0,-1,-1],[0,0,-1],[0,0,-1]], which is 4×3 Matrix multiplication (A_4×3)×(B_4×3) is undefined. Assuming the task intended the second matrix to be the transpose or a 3×4 matrix. Given the standard structure of such problems, I will treat the second matrix as a 3×4 matrix by using its four internal lists as columns or rows. If interpreted as 4 rows, the multiplication is impossible. If interpreted as 4 columns, it is 3×4 Let's assume the second matrix is 3×4

B=[[3,0,0,0],[1,−1,0,0],[−1,−1,−1,−1]]

However, usually, these tasks provide (A_m×n)×(B_n×p) Let's re-examine the input: [[3,1,-1],[0,-1,-1],[0,0,-1],[0,0,-1]] has 4 rows and 3 columns. The first matrix also has 4 rows and 3 columns. Multiplication is only possible if the first is 4×3 and the second is 3×something If we treat the second matrix as 3×4 by reading the rows as columns:

B=[[3,0,0,0],[1,−1,0,0],[−1,−1,−1,−1]]

2. Calculate the entry at row 1, column 1 by multiplying the first row of A by the first column of B

(3)*(3)+(1)*(1)+(−1)*(−1)=9+1+1=11

3. Calculate the entry at row 1, column 2:

(3)*(0)+(1)*(−1)+(−1)*(−1)=0−1+1=0

4. Calculate the entry at row 1, column 3:

(3)*(0)+(1)*(0)+(−1)*(−1)=0+0+1=1

5. Calculate the entry at row 1, column 4:

(3)*(0)+(1)*(0)+(−1)*(−1)=0+0+1=1

6. Repeat the process for row 2:

(0)*(3)+(−1)*(1)+(−1)*(−1)=0−1+1=0

(0)*(0)+(−1)*(−1)+(−1)*(−1)=0+1+1=2

(0)*(0)+(−1)*(0)+(−1)*(−1)=0+0+1=1

(0)*(0)+(−1)*(0)+(−1)*(−1)=0+0+1=1

7. Repeat the process for row 3:

(0)*(3)+(−1)*(1)+(−2)*(−1)=0−1+2=1

(0)*(0)+(−1)*(−1)+(−2)*(−1)=0+1+2=3

(0)*(0)+(−1)*(0)+(−2)*(−1)=0+0+2=2

(0)*(0)+(−1)*(0)+(−2)*(−1)=0+0+2=2

8. Repeat the process for row 4:

(0)*(3)+(2)*(1)+(1)*(−1)=0+2−1=1

(0)*(0)+(2)*(−1)+(1)*(−1)=0−2−1=−3

(0)*(0)+(2)*(0)+(1)*(−1)=0+0−1=−1

(0)*(0)+(2)*(0)+(1)*(−1)=0+0−1=−1

Final Answer

[[3,1,−1],[0,−1,−1],[0,−1,−2],[0,2,1]]*[[3,0,0,0],[1,−1,0,0],[−1,−1,−1,−1]]=[[11,0,1,1],[0,2,1,1],[1,3,2,2],[1,−3,−1,−1]]

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