00:01
Hi there in this question.
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We are given with certain statements and we have to state whether those are true or false.
00:06
So the first one is for every square matrix a, if a has a zero entry, then a is not invertible.
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Second one is if a is square matrix such that a x is equal to zero has non -trivial solution, then determinant of a is equal to 0.
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The third one is whenever a and b and b are invertible and by in matrices, we have a b whole inverse is equal to a inverse b inverse.
00:30
And now the fourth one is for all n by n matrix a, determinant of a power 5 is equal to determinant of a whole power 5.
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And the last one is for all square matrices of any order, determinant of 2a is equal to 2 multiplied by determinant of a.
00:47
So let's see whether these are true or false.
00:51
So the first statement is for every square matrix, say if a has a zero entry, then a is not invertible.
00:57
The first statement for this first statement consider the matrix 1 .001.
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This is the identity matrix and determinant of this matrix is determinant of this matrix, that is 1 .001.
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This matrix is equal to 1 which is not equal to 0.
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That means this matrix 1 -0 -1 is invertible.
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Is invertible.
01:26
So we can say that the first statement is false.
01:30
So the first statement is false.
01:33
The statement a is a false statement using this counter example of identity matrix of the order 2 by 2.
01:44
Now we can see the second statement.
01:49
The second statement is that if a square matrix such that a x is equal to to 0 has non -trivial solution, then determinant of a is equal to 0.
01:59
So it's given that a.
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X is equal to 0.
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We have two possibilities.
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That is, if we have a system a .x is equal to 0, then either there will be the first possibility is that unique solution and that unique solution will be x is equal to 0.
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And in this case, in the unique solution case, we have determinate a, that is determinant a not equal to 0.
02:25
Determinant a not equal to 0.
02:28
And the second case is that there are infinitely many solutions.
02:33
Infinitely many solutions.
02:36
Infinitely many solutions.
02:39
And there, x equal to 0, there will be only one of those solutions.
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And in this case, we have determinant of a is equal to 0.
02:46
And the third case is that no solution.
02:49
There exist no solution.
02:51
And in that case also we have determinant of a is equal to 0.
02:56
So it's given that ax is equal to 0.
02:59
Here the system ax is equal to 0 have non -trivial solution.
03:02
That means solution other than x is equal to 0.
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So the system will fall under the second category that is infinitely many solutions.
03:10
So there the determinant of a will be 0.
03:12
So the given statement is true.
03:15
So the b statement is.
03:19
So that's about the b statement.
03:21
Now we'll move on to the third one.
03:23
The third one is saying that whenever a and b are invertible n or n by n matrices, we have a, b whole inverse is equal to a inverse b inverse.
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So let's check that.
03:36
Let's check that.
03:37
So we know that if a and b are invertible matrices, if a b inverse exists, if a b inverse exists, then a b inverse multiplied by a b should be equal to the identity matrix.
03:52
Then if a b inverse inverse exists, then if a b inverse inverse, then if a b inverse inverse, is equal to if a b whole inverse is equal to a inverse b inverse b inverse then we'll be having that a b whole inverse a b is equal to a inverse b inverse ab is equal to i but we cannot make we cannot make a conclusion on this statement the given statement we cannot make a conclusion on the statement because we cannot combine b inverse a or the commutativity of the matrices is not well defined always...