LA Lecture 5

Question 1

What is the determinant of a 2x2 matrix
A =
a b
c d?

Answer 1

ad - bc

Question 2

What is the determinant of a 3x3 matrix
A =
a b c
d e f
g h i

Answer 2

(aei + bfg + cdh) - (gec +hfa + idb)

Question 3

What is the determinant of a singular matrix?

Answer 3

0

Question 4

What happens to the determinant if two rows of a matrix are exchanged?

Answer 4

The determinant changes sign (sign reversal)

Question 5

What is the determinant of a matrix after elimination (EA)?

Answer 5

The determinant is the product of the pivots, which equals the determinant of the original matrix (det EA = det A).

a ( d - [c/a]b)

Question 6

What is the determinant of the n by n identity matrix?

Answer 6

1

Question 7

What happens to the determinant if a row is multiplied by a number
t?

Answer 7

The determinant is multiplied by t

Question 8

What is the determinant if two rows of a matrix are equal?

Answer 8

The determinant is 0.

Question 9

What happens to the determinant if a multiple of one row is subtracted from another row?

Answer 9

The determinant remains unchanged.

Question 10

What is the determinant of a matrix with a row of zeros?

Answer 10

0

Question 11

What is the determinant of a triangular matrix?

Answer 11

The product of the diagonal entries.

Question 12

What is the determinant of a diagonal matrix?

Answer 12

The product of the diagonal entries.

Question 13

What is the determinant of a singular matrix?

Answer 13

0

Question 14

What is the determinant of the product of two matrices AB?

Answer 14

det A x det B

Question 15

What is the determinant of the transpose of a matrix A^T?

Answer 15

The same as the determinant of A
( det A^T = det A)

Question 16

What is the determinant of a 2x2 matrix
A =
a b
c d

Answer 16

ad - bc

Question 17

How many terms are in the determinant of an n×n matrix using the Big Formula?

Answer 17

n! terms

Question 18

What is the cofactor Cij of a matrix A?

Answer 18

Cij = (-1)^i+j det Mij,

Question 19

What is the relationship between the determinant of a matrix and its pivots in LU decomposition?

Answer 19

det A = d1d2…dn

Question 20

What is the determinant of a triangular matrix

Answer 20

The product of the diagonal entries.

Question 21

What is the determinant of the identity matrix?

Answer 21

1

Question 22

What happens to the determinant if two rows of a matrix are exchanged?

Answer 22

The determinant changes sign (sign reversal).

Question 23

What is the formula for the inverse of a matrix A using cofactors?

Answer 23

A^-1 = C^T/ det A
where C is the cofactor matrix.

Question 24

What is the formula for xij in Cramer’s Rule?

Answer 24

x j = det Bj / det A

Question 25

What is the area of a parallelogram formed by vectors (a,b) and (c,d)?

Answer 25

the area is |ad-bc|

Question 26

What is the volume of a box (parallelepiped) formed by the columns of a matrix A?

Answer 26

the volume is |det A|

Question 27

What is the cross product of two vectors u and v?

Answer 27

The cross product u×v is a vector perpendicular to both u and v

Question 28

What is the relationship between v×u and u×v?

Answer 28

v x u = - (u x v)

Question 29

what is the area of a triangle?

Answer 29

area = 1/2 determinant

Question 30

What is the cross product of i=(1,0,0) and j=(0,1,0)?

Answer 30

i x j = k, where k = (0,0,1)

Question 31

What is the determinant of a matrix A if its rows (or columns) are the sides of a box?

Answer 31

The determinant detA gives the volume of the box.

Question 32

What is the volume of a box formed by the columns of a 3×3 matrix A?

Answer 32

volume is |det A|

Question 33

cross product of u = (1,1,1) and v = (1,1,2)

Answer 33

(1, -1, 0)

Question 34

What is the determinant of a matrix A if its rows are linearly dependent?

Answer 34

If the rows are linearly dependent, det A = 0