Let C and D be n x n matrices. Which of the following...

Let C and D be n x n matrices. Which of the following statements are true? Statement A: det(C)det(D) = det(DC) Statement B: det(kC) = kdet(C) Both A and B A only B ony Neither A nor B

Transcript

00:01 In this problem we have given c and d b n cross n matrix.

00:10 So we have given two matrices b and d and the dimension is n cross n and here we have given two statement the statement first is determinant of c into determinant of d equal determinant of d c and the statement second is second statement is here determinant of k c equal k f determinant of so we have to find here we have to find here which of the following statement this is a and b not a 1 2 is true so first one here we have given determinant of c into determinant of d that is equal to determinant of dc so this is an standard result of determinant so this is the property of the determinant of a cross a matrix so it also can be written as determinant of d c equal determinant of d into determinant of c so you can take either of our determinant of c first or d so this is an important property of terminant of product of matrices so this statement is true statement one is statement a is true talking about statement two is statement b so b is here determinant of ac equal k of terminate so here this is a false statement the correct will be determinant of a equal k power n where n is the order of matrix c equal into determinant of c this is also a property now we will prove we will give a counter example here to so this is a false statement so let's a matrix here let's a matrix equal 1 2 and 1 4 first we will find determinant of so this is 1 2, 1 4, that is 4 minus 2.

02:54 So this is 2.

02:56 Now let's take a value k equal 2 and put here find k into c.

03:06 So k into c equal multiplication of matrix y 2.

03:11 So this is 1 2 ,14 that is k c equal 2 4 and 2.

03:19 So this is matrix kc.

03:21 Now we will find the determinant of this matrix kc.

03:26 So the determinant of kc that is determinant 2 4 to 8.

03:34 This is 16 minus 8.

03:39 That is 8.

03:41 So here we can see that the determinant of c is 2...

Question

Let C and D be n x n matrices. Which of the following statements are true? Statement A: det(C)det(D) = det(DC) Statement B: det(kC) = kdet(C) Both A and B A only B ony Neither A nor B

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