Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
