Answer :
To find the inverse of a matrix using the Gauss-Jordan method (also known as row-reduction), we can follow these steps:
1. Form the augmented matrix [A | I], where A is the original matrix and I is the identity matrix of the same size.
2. Perform row operations to transform the matrix A into the identity matrix I.
3. The resulting matrix on the right-hand side of the augmented matrix will be the inverse [tex]\( A^{-1} \)[/tex].
Let's perform the Gauss-Jordan method on each of the given matrices.
### Part (a)
Matrix [tex]\( \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 2 & 5 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations:
- Row 2: [tex]\(R_2 \rightarrow R_2 - 2R_1 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 0 & 1 & -2 & 1 \end{array}\right) \][/tex]
- Row 1: [tex]\(R_1 \rightarrow R_1 - 2R_2 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & 0 & 5 & -2 \\ 0 & 1 & -2 & 1 \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} 5 & -2 \\ -2 & 1 \end{pmatrix} \][/tex]
### Part (b)
Matrix [tex]\( \begin{pmatrix} -3 & -5 \\ 6 & 8 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{cc|cc} -3 & -5 & 1 & 0 \\ 6 & 8 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations:
- Row 1: [tex]\(R_1 \rightarrow -\frac{1}{3} R_1\)[/tex]:
[tex]\[ \left(\begin{array}{cc|cc} 1 & \frac{5}{3} & -\frac{1}{3} & 0 \\ 6 & 8 & 0 & 1 \end{array}\right) \][/tex]
- Row 2: [tex]\(R_2 \rightarrow R_2 - 6R_1 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & \frac{5}{3} & -\frac{1}{3} & 0 \\ 0 & -2 & 2 & 1 \end{array}\right) \][/tex]
- Row 2: [tex]\(R_2 \rightarrow -\frac{1}{2} R_2\)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & \frac{5}{3} & -\frac{1}{3} & 0 \\ 0 & 1 & -1 & -\frac{1}{2} \end{array}\right) \][/tex]
- Row 1: [tex]\(R_1 \rightarrow R_1 - \frac{5}{3}R_2 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & 0 & \frac{4}{3} & \frac{5}{6} \\ 0 & 1 & -1 & -\frac{1}{2} \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} \frac{4}{3} & \frac{5}{6} \\ -1 & -\frac{1}{2} \end{pmatrix} \][/tex]
### Part (c)
Matrix [tex]\( \begin{pmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{ccc|ccc} 1 & -2 & 3 & 1 & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ -2 & 2 & 1 & 0 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations to transform the left matrix to identity:
[tex]\[ \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & -9 & 8 & -5 \\ 0 & 1 & 0 & -8 & 7 & -4 \\ 0 & 0 & 1 & -2 & 2 & -1 \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} -9 & 8 & -5 \\ -8 & 7 & -4 \\ -2 & 2 & -1 \end{pmatrix} \][/tex]
### Part (d)
Matrix [tex]\( \begin{pmatrix} 3 & 2 & 6 \\ 1 & 1 & 3 \\ 2 & 3 & 4 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{ccc|ccc} 3 & 2 & 6 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 & 1 & 0 \\ 2 & 3 & 4 & 0 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations to transform the left matrix to identity:
[tex]\[ \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & -2 & 0 \\ 0 & 1 & 0 & -\frac{2}{5} & 0 & \frac{3}{5} \\ 0 & 0 & 1 & -\frac{1}{5} & 1 & -\frac{1}{5} \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} 1 & -2 & 0 \\ -\frac{2}{5} & 0 & \frac{3}{5} \\ -\frac{1}{5} & 1 & -\frac{1}{5} \end{pmatrix} \][/tex]
These are the inverses of the provided matrices using the Gauss-Jordan method.
1. Form the augmented matrix [A | I], where A is the original matrix and I is the identity matrix of the same size.
2. Perform row operations to transform the matrix A into the identity matrix I.
3. The resulting matrix on the right-hand side of the augmented matrix will be the inverse [tex]\( A^{-1} \)[/tex].
Let's perform the Gauss-Jordan method on each of the given matrices.
### Part (a)
Matrix [tex]\( \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 2 & 5 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations:
- Row 2: [tex]\(R_2 \rightarrow R_2 - 2R_1 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 0 & 1 & -2 & 1 \end{array}\right) \][/tex]
- Row 1: [tex]\(R_1 \rightarrow R_1 - 2R_2 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & 0 & 5 & -2 \\ 0 & 1 & -2 & 1 \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} 5 & -2 \\ -2 & 1 \end{pmatrix} \][/tex]
### Part (b)
Matrix [tex]\( \begin{pmatrix} -3 & -5 \\ 6 & 8 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{cc|cc} -3 & -5 & 1 & 0 \\ 6 & 8 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations:
- Row 1: [tex]\(R_1 \rightarrow -\frac{1}{3} R_1\)[/tex]:
[tex]\[ \left(\begin{array}{cc|cc} 1 & \frac{5}{3} & -\frac{1}{3} & 0 \\ 6 & 8 & 0 & 1 \end{array}\right) \][/tex]
- Row 2: [tex]\(R_2 \rightarrow R_2 - 6R_1 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & \frac{5}{3} & -\frac{1}{3} & 0 \\ 0 & -2 & 2 & 1 \end{array}\right) \][/tex]
- Row 2: [tex]\(R_2 \rightarrow -\frac{1}{2} R_2\)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & \frac{5}{3} & -\frac{1}{3} & 0 \\ 0 & 1 & -1 & -\frac{1}{2} \end{array}\right) \][/tex]
- Row 1: [tex]\(R_1 \rightarrow R_1 - \frac{5}{3}R_2 \)[/tex]
[tex]\[ \left(\begin{array}{cc|cc} 1 & 0 & \frac{4}{3} & \frac{5}{6} \\ 0 & 1 & -1 & -\frac{1}{2} \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} \frac{4}{3} & \frac{5}{6} \\ -1 & -\frac{1}{2} \end{pmatrix} \][/tex]
### Part (c)
Matrix [tex]\( \begin{pmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{ccc|ccc} 1 & -2 & 3 & 1 & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ -2 & 2 & 1 & 0 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations to transform the left matrix to identity:
[tex]\[ \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & -9 & 8 & -5 \\ 0 & 1 & 0 & -8 & 7 & -4 \\ 0 & 0 & 1 & -2 & 2 & -1 \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} -9 & 8 & -5 \\ -8 & 7 & -4 \\ -2 & 2 & -1 \end{pmatrix} \][/tex]
### Part (d)
Matrix [tex]\( \begin{pmatrix} 3 & 2 & 6 \\ 1 & 1 & 3 \\ 2 & 3 & 4 \end{pmatrix} \)[/tex]:
1. Set up the augmented matrix:
[tex]\[ \left(\begin{array}{ccc|ccc} 3 & 2 & 6 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 & 1 & 0 \\ 2 & 3 & 4 & 0 & 0 & 1 \end{array}\right) \][/tex]
2. Perform row operations to transform the left matrix to identity:
[tex]\[ \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & -2 & 0 \\ 0 & 1 & 0 & -\frac{2}{5} & 0 & \frac{3}{5} \\ 0 & 0 & 1 & -\frac{1}{5} & 1 & -\frac{1}{5} \end{array}\right) \][/tex]
3. The inverse is:
[tex]\[ \begin{pmatrix} 1 & -2 & 0 \\ -\frac{2}{5} & 0 & \frac{3}{5} \\ -\frac{1}{5} & 1 & -\frac{1}{5} \end{pmatrix} \][/tex]
These are the inverses of the provided matrices using the Gauss-Jordan method.
