Find the Inverse of the Resulting Matrix [[4],[5],[7]]*[[4],[6],[2]]

Problem

([[4],[5],[7]]*[[4,6,2]])(−1)

Solution

1. Multiply the given matrices to find the resulting matrix A Since we are multiplying a 3×1 matrix by a 1×3 matrix, the result is a 3×3 matrix.

A=[[4*(4),4*(6),4*(2)],[5*(4),5*(6),5*(2)],[7*(4),7*(6),7*(2)]]

A=[[16,24,8],[20,30,10],[28,42,14]]

2. Determine the rank of the resulting matrix. Notice that each row is a scalar multiple of the vector [[4,6,2]] Specifically, the second row is 1.25 times the first row, and the third row is 1.75 times the first row.

(Row_2)=5/4*(Row_1)

(Row_3)=7/4*(Row_1)

3. Calculate the determinant of the matrix. Because the rows are linearly dependent (the rows are multiples of each other), the matrix is singular.

det(A)=0

4. Conclude that the inverse does not exist. A matrix is invertible if and only if its determinant is non-zero. Since the determinant is zero, the matrix is non-invertible (singular).

Final Answer

([[4],[5],[7]]*[[4,6,2]])(−1)=Undefined (Matrix is singular)

Want more problems? Check here!