5. Find the cofactor matrix for A = and use it to find A- 6. Find the cofactor matrix for and use it to generate the formula for a 2-by-2 inverse.
5. Find the cofactor matrix for A = and use it to find A- 6. Find the cofactor matrix for and use it to generate the formula for a 2-by-2 inverse.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![In section 5.3 we used Cramer's Rule to find the solution to Ax = b for the specific variable x.
To do this, we replaced the j-th column of the coefficient matrix A with the vector b to get the
matrix Bj. Then,
det(B)
X, = det(A)
We can use Cramer's Rule to find the inverse of a matrix one entry at a time:
Since AA-1 = 1, if we multiply A by the first column of A-1 we should get the first column of I.
Likewise, if we multiply A by any other column of A-1 we get the corresponding column of I.
|1 3 -1]
1. Let A =|-1 2 1. Let bị = 0
(i.e. the first column of I) and solve Ax = bị using
-2 -2 3
Cramer's Rule (to get the first column of A-1). Note that this will require using Cramer's
Rule three times, once for each component.
2. Now let bz = 1 and solve Ax = b2.
3. Solve Ax = bz one last time with bz =
4. Use the results of the first three questions to write A-1.
In general, we can find the ij-th entry of A- by using Cramer's Rule, replacing the i-th column of
A with the j-th column of 1. Then, the determinant of this matrix can be found using cofactor
expansion down that i-th column. The only nonzero term of that expansion will be in the j-th row.
This gives us:
det(B1) _ (–1)**/ det(M1)
det(A)
det(A)
where Mji is the "minor" matrix obtained by removing the j-th row and i-th column of A. Notice
in particular that, because of our earlier construction, the row and column are flipped! The ij-th
entry of A- depends on the cofactor of the ji-th entry of A.
We can simplify the notation by creating a "cofactor matrix" C, whose ij-th entry is the cofactor
of the ij-th entry of A. Then:
CT
A-1
det(A)
5. Find the cofactor matrix for A =
and use it to find A-1
6. Find the cofactor matrix for a 2I
and use it to generate the formula for a 2-by-2 inverse.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa658c71c-1a2c-47f6-b23a-f861466702f5%2F7a9f1b91-d85e-4ac5-93a0-b26fea6048ef%2F4h439te_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In section 5.3 we used Cramer's Rule to find the solution to Ax = b for the specific variable x.
To do this, we replaced the j-th column of the coefficient matrix A with the vector b to get the
matrix Bj. Then,
det(B)
X, = det(A)
We can use Cramer's Rule to find the inverse of a matrix one entry at a time:
Since AA-1 = 1, if we multiply A by the first column of A-1 we should get the first column of I.
Likewise, if we multiply A by any other column of A-1 we get the corresponding column of I.
|1 3 -1]
1. Let A =|-1 2 1. Let bị = 0
(i.e. the first column of I) and solve Ax = bị using
-2 -2 3
Cramer's Rule (to get the first column of A-1). Note that this will require using Cramer's
Rule three times, once for each component.
2. Now let bz = 1 and solve Ax = b2.
3. Solve Ax = bz one last time with bz =
4. Use the results of the first three questions to write A-1.
In general, we can find the ij-th entry of A- by using Cramer's Rule, replacing the i-th column of
A with the j-th column of 1. Then, the determinant of this matrix can be found using cofactor
expansion down that i-th column. The only nonzero term of that expansion will be in the j-th row.
This gives us:
det(B1) _ (–1)**/ det(M1)
det(A)
det(A)
where Mji is the "minor" matrix obtained by removing the j-th row and i-th column of A. Notice
in particular that, because of our earlier construction, the row and column are flipped! The ij-th
entry of A- depends on the cofactor of the ji-th entry of A.
We can simplify the notation by creating a "cofactor matrix" C, whose ij-th entry is the cofactor
of the ij-th entry of A. Then:
CT
A-1
det(A)
5. Find the cofactor matrix for A =
and use it to find A-1
6. Find the cofactor matrix for a 2I
and use it to generate the formula for a 2-by-2 inverse.
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