If two matrices are row-equivalent (that is, if the two matrices are related by
ID: 3136358 • Letter: I
Question
If two matrices are row-equivalent (that is, if the two matrices are related by elementary row operations), then either both are invertible or neither is invertible. 9 127 3 7 5 4153 8 172 9 8 7 6 5 4 173 Decide whether the matrix M invertible by performing elementary row operations to obtain a matrix which is either clearly invertible or clearly not. (The matrix you obtain should either obviously have zero determinant or obviously have non-zero determinant, without having to resort to a calculation beyond the row operations.)Explanation / Answer
Let us perform the following row operations on M.
Multiply the 1st row by 1/9
Add -5 times the 1st row to the 2nd row
Add -172 times the 1st row to the 3rd row
Add -6 times the 1st row to the 4th row
Multiply the 2nd row by -9/599
Add 21763/9 times the 2nd row to the 3rd row
Add 239/3 times the 2nd row to the 4th row
Multiply the 3rd row by -599/3323018
Add 107308/599 times the 3rd row to the 4th row
Multiply the 4th row by 1661509/286327304
Then M changes to
1
127/9
1/3
7/9
0
1
-1362/599
-37/599
0
0
1
82705/1661509
0
0
0
1
This is an upper triangular matrix. We know that the determinant of an upper triangular matrix is the product of the diagonal entries. Since none of the diagonal entries is 0, the determinant of this matrix is non-zero. Hence this matrix is invertible. Therefore, M, which is row equivalent to this matrix, is also invertible.
1
127/9
1/3
7/9
0
1
-1362/599
-37/599
0
0
1
82705/1661509
0
0
0
1