Write the 2x2 matrix of X= -3 1 as a product X= E1E2E3 of elementary matrices
-9 0
To write the 2x2 matrix X as a product of elementary matrices, we need to perform a series of elementary row operations on the identity matrix until it becomes X.
Let's start with the identity matrix: I = [1 0; 0 1].
Step 1: Perform a row addition operation
To change the first row of the identity matrix to match the first row of matrix X, we can subtract 3 times the first row from the second row:
E1 = [1 0; -3 1]
Multiply E1 with the identity matrix:
E1 * I = [1 0; -3 1] * [1 0; 0 1] = [1 0; -3 1]
Step 2: Perform a scalar multiplication operation
To change the second row of the resulting matrix from the previous step, we need to multiply it by -9/3 = -3:
E2 = [1 0; 0 -3]
Multiply E2 with the matrix obtained from the previous step:
E2 * [1 0; -3 1] = [1 0; 0 -3] * [1 0; -3 1] = [1 0; 0 3]
Step 3: Perform a row addition operation
To change the second row of the resulting matrix from step 2, we can add 3 times the second row to the first row:
E3 = [1 3; 0 1]
Multiply E3 with the matrix obtained from the previous step:
E3 * [1 0; 0 3] = [1 3; 0 1] * [1 0; 0 3] = [1 3; 0 3]
By multiplying the three elementary matrices together, we obtain:
E1 * E2 * E3 = [1 0; -3 1] * [1 0; 0 -3] * [1 3; 0 1] = [1 3; 0 3]
Therefore, the 2x2 matrix X can be represented as a product of elementary matrices as follows:
X = E1 * E2 * E3 = [1 3; 0 3]