Transform the matrix A=[5,−3;1,1], into a matrix B using the similarity transformation (C^−1)AC, with matrix
C=[3,1;1,1].
B=(C^−1)AC = ?
well, C^-1 is (1/2)[1,-1;-1,3]
so just crank it out. You can check your answer here. Note the notation for matrices.
http://www.wolframalpha.com/input/?i=inverse+{{3,1},{1,1}}
To find the matrix B using the similarity transformation (C^−1)AC, we need to follow these steps:
Step 1: Calculate the inverse of matrix C.
To find the inverse of a 2x2 matrix C, we can use the following formula:
C^−1 = (1/det(C)) * adj(C)
where det(C) represents the determinant of matrix C and adj(C) represents the adjugate of matrix C.
Let's calculate the determinant of C:
det(C) = (3*1) - (1*1) = 3 - 1 = 2
Now, let's find the adjugate of C:
adj(C) = [d, -b; -c, a]
where a, b, c, d represent the entries of matrix C.
adj(C) = [1, -1; -1, 3]
Finally, we can calculate the inverse of C:
C^−1 = (1/2) * [1, -1; -1, 3] = [1/2, -1/2; -1/2, 3/2]
Step 2: Multiply (C^−1) with matrix A and then multiply the result with C.
B = (C^−1)AC
Let's perform the calculation:
(C^−1) = [1/2, -1/2; -1/2, 3/2]
A = [5, -3; 1, 1]
Multiply (C^−1) with A:
(C^−1)A = [1/2, -1/2; -1/2, 3/2] * [5, -3; 1, 1] = [8, -8; -1, 2]
Now multiply the result with C:
B = (C^−1)AC = [8, -8; -1, 2] * [3, 1; 1, 1] = [16, 0; -5, -7]
Therefore, the matrix B = [16, 0; -5, -7]