Select all statements below which are true for all...

Select all statements below which are true for all invertible n × n matrices A and B A. A^3 B^6 is invertible B. A + I_n is invertible C. AB = BA D. (A + B)(A - B) = A^2 - B^2 E. (A + A^-1)^8 = A^8 + A^-8 F. (I_n - A)(I_n + A) = I_n - A^2

Transcript

00:01 Hi, today we are solving the question in which we have to find out given statement which is true.

00:10 So here firstly we are given that in the statement a plus b is invertible.

00:16 So here let us consider this statement.

00:20 So multiply both sides by a.

00:24 So that is the sum of two invertible matrices.

00:30 So the sum of invertible matrices is always invertible.

00:40 So a plus b is invertible.

00:42 This statement is true.

00:44 Now consider b statement that is a plus a inverse whole raise to the power 8 is equals to a raise to the power 8 plus a raise to the power minus 8.

00:58 So here using binomial expansion we get a plus a raise to the power minus 8.

01:06 Plus a inverse whole raise to the power 8 is equal to a raised to the power 8 plus 8 into a raise to the power 8 into a inverse whole inverse plus 32 into a raise to the power 6 into a inverse whole square till a inverse raised to the power 8.

01:37 Now to simplify, we'll use the rule that a raised to the power m and m whole n is equal to a raised to the power m into n.

01:57 And a raised to the power m into a raise to the power n is equals to a raise to the power m plus n.

02:07 So from here, we get the statement as so it is equals to a raise to the power 8 plus 8 into a raise to the power 6 plus 32 into a raise to the power 6 till a raise to the bar minus 8.

02:29 So from here we can see that a plus a inverse whole raise to the power 8 is not equal to a race to the power 8 plus a raise to the power minus 8.

02:40 So this given statement is false.

02:44 Now similarly let us consider the c statement that is a into b into a inverse is equal to b.

02:55 So from here we can see that if we multiply both sides by a.

03:02 So we get a b, a inverse into a is equal to b into a...

Question

Select all statements below which are true for all invertible n × n matrices A and B A. A^3 B^6 is invertible B. A + I_n is invertible C. AB = BA D. (A + B)(A - B) = A^2 - B^2 E. (A + A^-1)^8 = A^8 + A^-8 F. (I_n - A)(I_n + A) = I_n - A^2

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