Answer :
The elementary matrix E that satisfies the equation ED = C is given by: E = ⎣⎡100001000010⎦⎤
To find an elementary matrix E that satisfies the equation ED = C, we can use elementary row operations to transform the matrix D into the matrix C. Each elementary row operation can be represented by an elementary matrix, and the product of these elementary matrices will give us the desired E.
Performing row operations on D to transform it into C:
1. Multiply the first row of D by -3 and add it to the second row: R2 = R2 - 3R1.
2. Swap the second and third rows of the resulting matrix.
The resulting matrix will be C.
Let's write down the elementary matrices corresponding to these operations:
E1 = ⎣⎡1−3⋅000−1⋅0⎦⎤ = ⎣⎡1−3⋅0000−1⋅0⎦⎤ = ⎣⎡1−3⋅0000−1⋅0⎦⎤ = ⎣⎡1−3⋅0000−1⋅0⎦⎤ = ⎣⎡1−3⋅0000−1⋅0⎦⎤ = ⎣⎡1−3⋅0000−1⋅0⎦⎤ = ⎣⎡1000⎦⎤
E2 = ⎣⎡100001000010⎦⎤
Thus, the elementary matrix E that satisfies the equation ED = C is the product of E2 and E1:
E = E2 x E1 = ⎣⎡100001000010⎦⎤ x ⎣⎡1000⎦⎤
E = ⎣⎡100001000010⎦⎤ x ⎣⎡1000⎦⎤ = ⎣⎡100001000010⎦⎤
Therefore, the elementary matrix E that satisfies the equation ED = C is given by:
E = ⎣⎡100001000010⎦⎤
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