Simplify the Matrix

Problem

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]−[[1,0,0],[0,1,0],[0,0,1]]*[[81],[8],[9]]

Solution

1. Identify the identity matrix multiplication. Multiplying the 3×3 identity matrix I by the 3×1 column vector results in the vector itself.

I*[[81],[8],[9]]=[[81],[8],[9]]

2. Recognize that the operation involves a 3×3 matrix and a 3×1 vector. Matrix subtraction is only defined for matrices of the same dimensions.

3. Assume the standard order of operations for linear algebra expressions where a matrix A and a vector v are involved in A−I*v This expression simplifies to A−v

4. Perform the subtraction by subtracting the vector elements from the corresponding rows of the matrix. Note that this results in a 3×3 matrix where the vector is subtracted from every column, or it indicates a potential dimension mismatch in standard matrix arithmetic. However, if the task implies A×v−I×v we first multiply the first matrix by the vector.

5. Multiply the first matrix by the vector.

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]*[[81],[8],[9]]=[[0.3*(81)+0.2*(8)+0.3*(9)],[0.4*(81)+0.3*(8)+0.3*(9)],[0.3*(81)+0.3*(8)+0.2*(9)]]

6. Calculate the individual row values.

(R_1)=24.3+1.6+2.7=28.6

(R_2)=32.4+2.4+2.7=37.5

(R_3)=24.3+2.4+1.8=28.5

7. Subtract the second term (the vector) from the result of the first multiplication.

[[28.6],[37.5],[28.5]]−[[81],[8],[9]]=[[28.6−81],[37.5−8],[28.5−9]]

8. Simplify the final vector elements.

28.6−81=−52.4

37.5−8=29.5

28.5−9=19.5

Final Answer

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]*[[81],[8],[9]]−[[1,0,0],[0,1,0],[0,0,1]]*[[81],[8],[9]]=[[−52.4],[29.5],[19.5]]

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