Get personalized answers to your unique questions on IDNLearn.com. Join our community to receive prompt, thorough responses from knowledgeable experts.
Sagot :
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is committed to providing the best answers. Thank you for visiting, and see you next time for more solutions.
