Answer:
To find the inverse of the matrix A, we will use the row reduction method. We will augment matrix A with the identity matrix I and perform row operations until A is transformed into the identity matrix. The resulting matrix on the right side will be the inverse of A.
Explanation:
Augment the matrix A with the identity matrix I:
[ 1 0 -3 | 1 0 0 ]
[ 3 1 -4 | 0 1 0 ]
[ 4 2 -4 | 0 0 1 ]
Perform row operations to transform the left side of the augmented matrix into the identity matrix:
R2 = R2 - 3R1
R3 = R3 - 4R1
[ 1 0 -3 | 1 0 0 ]
[ 0 1 5 | -3 1 0 ]
[ 0 2 8 | -4 0 1 ]
Perform row operations to further transform the left side of the augmented matrix into the identity matrix:
R3 = R3 - 2R2
[ 1 0 -3 | 1 0 0 ]
[ 0 1 5 | -3 1 0 ]
[ 0 0 -2 | 2 -2 1 ]
Multiply the third row by -1/2 to make the pivot element of the third row equal to 1:
R3 = (-1/2) * R3
[ 1 0 -3 | 1 0 0 ]
[ 0 1 5 | -3 1 0 ]
[ 0 0 1 | -1 1 -1/2 ]
Perform row operations to further transform the left side of the augmented matrix into the identity matrix:
R1 = R1 + 3R3
R2 = R2 - 5R3
[ 1 0 0 | 2 0 3/2 ]
[ 0 1 0 | 2 -4 5/2 ]
[ 0 0 1 | -1 1 -1/2 ]
The resulting matrix on the right side of the augmented matrix is the inverse of matrix A:
[ 2 0 3/2 ]
[ 2 -4 5/2 ]
[ -1 1 -1/2 ]
Therefore, the inverse of matrix A is:
[ 2 0 3/2 ]
[ 2 -4 5/2 ]
[ -1 1 -1/2 ]