Determinant Properties

Question 1

Whats the first axiom?

Answer 1

Det(I_n)=1 for all positive integer n

Question 2

Whats the second axiom?

Answer 2

Row interchange/swapping rows negates the sign of the determinant

Question 3

Whats the third axiom?

Answer 3

The determinant is a linear function in each row separately given that al the other rows stay the same

Question 4

Whats the fourth property

Answer 4

If A has two equal rows, detA=0

Question 5

Whats the fifth property

Answer 5

Adding/subtracting a multiple of one row into another row (aka, doing row replacement) does NOT change the determinant

Question 6

Whats the sixth property?

Answer 6

If A has a zero row, then the determinant is 0

Question 7

Whats the seventh property?

Answer 7

If A is a triangular matrix, then the determinant is the product of the diagonals

Question 8

Eighth property

Answer 8

det(AB)=det(A)*det(B)

Question 9

Ninth property

Answer 9

Det(A^T)=det(A)

Question 10

10th property

Answer 10

If A is invertible, then det(A^-1)=1/det(A)

Question 11

Property 2B

Answer 11

Interchanging two columns also negates the determinant

Question 12

Property 4B

Answer 12

If A has two equal columns, then det(A)=0

Question 13

Property 6B

Answer 13

If A has a zero column, then det(A)=0

Question 14

det(kA)=

Answer 14

(k^n)*det(A), where n is the dimension of the square matrix (n is 3 if 3x3)

Question 15

Det 2x2

Answer 15

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Question 16

How find determinant of 3x3 or higher

Answer 16

Do NOT scale rows, do row replacement to ref and keep note of the number of swaps and use that to invert at end. Once u get to ref, just multiply the diagonal

Question 17

Det(-A)=

Answer 17

(-1)^n*det(A)