Suppose p(λ) = λ(λ−1)(λ+2)(λ+1) is the characteristic polynomial of a matrix a. consider the statements:
i: a must be a 4 × 4 matrix.
ii: a can be diagonalized.
iii: a is invertible.
which of the statements i, ii, and iii are false?

 The false statement is

III. A is invertible.

Step-by-step explanation:  Given that the characteristic polynomial of a matrix A is

We are to select the statement that is FALSE.

The characteristic values of A are given by

.

I. The characteristic polynomial of A has degree 4, so the order of A is 4. That is, the matrix A must be a 4 × 4 matrix.

So, the statement I is true.

II. Since all the characteristic values of A are all distinct, so there will be 4 linearly independent characteristic vectors.

Therefore, there exists an invertible matrix P such that where D is daigonal with characteristic values as diagonal matrix.

This makes A diagonalizable.

So, the statement II is TRUE.  

III. We know that the determinant of a matrix is equal to the product of the characteristic values of the matrix.

Therefore,

This implies that A is non-invertible.

So, statement III is FALSE.

Thus, statement III is FALSE.

the value of x is 9 ⇒ answer b

step-by-step explanation:

* we can solve this problem using cosine and sine rule

- in δ abc

∵ ab = 36 , bc = 28 , ac = 16

- lets find the measure of angle b using the cosine rule

∵ cos b = [ab² + bc² - ac²]/2(ab)(bc)

∴ cos b = [36² + 28² - 16²]/2(36)(28)

∴ cos b = 19/21

∴ m∠b = cos^-1(19/21) ≅ 25°

- lets find the measure of angle a using the sine rule

∵ sina/bc = sinb/ac = sinc/ab

∴ sina/28 = sin25°/16 ⇒ by using cross multiplication

∴ sina = 28 × sin25°/16

∴ sina = 0.7396

∴ m∠a = sin^-1(0.7396) ≅ 48°

- from the figure m∠abd = m∠dbc

∵ m∠abc = 25°

∴ m∠abd = 25°/2 = 12.5°

- in δabd

∵ m∠a = 48° , m∠abd = 12.5°

∵ the sum of the measures of the interior angles of any δ is 180°

∴ m∠adb = 180° - (48° + 12.5°) = 119.5°

∵ ab = 36

∵ ad = x

- use the sine rule to find x

∵ sin∠adb/ab = sin∠abd/x

∴ sin119.5°/36 = sin12.5°/x ⇒ by using cross multiplication

∴ x = 36 × sin12.5°/sin119.5° = 8.95 ≅ 9

* the value of x is 9

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step-by-step explanation: