Use the fact that |c A|=c^n|A| to evaluate the determinant of the n ×n matrix. A=[ 2 -4

Use the fact that |c A|=c^n|A| to evaluate the determinant...

Key Concepts

Scalar Multiplication Property for Determinants

The property |cA| = c^n |A| states that if every entry in an n x n matrix is multiplied by a scalar c, then the determinant of the resulting matrix is equal to c raised to the power n times the determinant of the original matrix. This property allows for simplified computation of determinants when a common factor is present across the matrix.

Determinant

The determinant of a square matrix is a scalar value that encapsulates important properties of the matrix, such as whether the matrix is invertible and how it scales volumes or areas in the corresponding space. It can be computed using various techniques including cofactor expansion and row reduction.

Transcript

00:01 Hello students we are given a matrix and we need to find the determinant of that particular matrix okay so we can write the matrix a in the form of see how we will write it two is nothing but twice of one okay minus four can be written as twice of minus two six is nothing but twice of three minus four is nothing but again two times of minus two six is nothing but twice of three again six twice of three minus eight is nothing but twice of minus four and same twice of minus four and and 10 is nothing but twice of 5.

00:33 Now, what we will have to do, we just take 2 as common from the matrix, okay? now, twice of, we can write it as 1, minus 2, 3, minus 2, 3, minus 4, and 3 minus 4, and last term is 5.

00:49 Now, what we need to do is we just need to calculate it as a determinant, so we can write it as determinant of.

00:55 See, how we will write it? determinate of twice of, okay, students, and the base, particular matrix of 1 minus 2 3 and the second column is minus 2 3 minus 4 and another column is 3 minus 5 okay and just close it and it's all about our determinant and now how we will apply the formula we are given that we need to apply the c times of a that a matrix the determinant of c a is equal to c to the power n times the determinant of a okay where n is just a order of the matrix here we are we can say that it is a three cross three matrix it means n is nothing but three so what we can do is we can just evaluate it just like that so c to the power n is nothing but two to the power three okay after that it will be comes out as the determinant of one minus two three minus two three minus four and five okay now we will expand it determinant of a is equal to two cube is nothing but eight okay and times one times of see three fives are 15 minus minus four times minus four is nothing but 16 okay plus 16 and minus of minus of minus two times of see how will calculate minus two times of five is nothing but minus 10 minus three times of minus four is nothing but minus 12 so this will be minus minus minus plus and 12 okay and plus three times of see minus 2 times of minus 4 is nothing but 8 minus 3 times of 3 is nothing but 9.

02:30 Okay, so what we can do is we can just write it as 8 times of 15 minus 16 is nothing but minus 1.

02:38 Okay and minus minus plus and twice of 12 minus 10 is nothing but 2...

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