Matrices

Question 1

What is a matrix?

Answer 1

A rectangular array of numbers arranged in rows and columns, representing linear transformations.

Question 2

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

Answer 2

det(A) = ad − bc

Question 3

What does the determinant signify?

Answer 3

It indicates if a matrix is invertible (det ≠ 0) and the scaling factor of the transformation.

Question 4

What is the formula for the inverse of a 2x2 matrix A = [[a, b], [c, d]]?

Answer 4

A⁻¹ = (1/det(A)) * [[d, -b], [-c, a]], provided det(A) ≠ 0.

Question 5

What does matrix multiplication represent?

Answer 5

Combining two linear transformations sequentially.

Question 6

Is matrix multiplication commutative?

Answer 6

No, A ⋅ B ≠ B ⋅ A in general.

Question 7

How do you calculate the transpose of a matrix A?

Answer 7

Flip rows and columns: Aᵀ

Question 8

What is the rank of a matrix?

Answer 8

The rank is the number of linearly independent rows or columns.

Question 9

How do you find the trace of a matrix A?

Answer 9

Sum of the diagonal elements.

Question 10

what is the row echelon form of a matric and which properties does a matrix need to have to be in row echelon form?

Answer 10

1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1 (does not apply to all definitions)
2. All zeros rows are at the bottom of the matrix
3. In any two successive rows that do not consist rows that do not consist entirely of 0s, the leading 1 in the lower row occurs further to the right than the leading 1 in the higher row

• it is not unique

Question 11

What is reduced row echelon form

Answer 11

Each column containing a leading 1 has zeros in all its other entries

• it is unique

Question 12

Can every matrix be turned into row echelon form?

Answer 12

Yes, through gaussian elimination