Matrices And Determinants

Question 1

What is a matrix?

Answer 1

A matrix is a rectangular array of numbers arranged in rows and columns.

Example:
Matrix A = (egin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 end{pmatrix})

Question 2

What is a row matrix?

Answer 2

A row matrix has only one row.

Example:
Row matrix R = (egin{pmatrix} 1 & 2 & 3 end{pmatrix})

Question 3

What is a column matrix?

Answer 3

A column matrix has only one column.

Example:
Column matrix C = (egin{pmatrix} 4 \ 5 \ 6 end{pmatrix})

Question 4

What is a square matrix?

Answer 4

A square matrix has the same number of rows and columns.

Example:
Square matrix S = (egin{pmatrix} 1 & 2 \ 3 & 4 end{pmatrix})

Question 5

How do you denote a matrix?

Answer 5

A matrix is usually denoted by a capital letter (e.g., A, B, C).

Question 6

What is matrix addition?

Answer 6

Adding corresponding elements of two matrices of the same dimension.

Question 7

What is the rule for matrix multiplication?

Answer 7

The number of columns in the first matrix must equal the number of rows in the second matrix.

Question 8

What is an identity matrix?

Answer 8

A square matrix with 1s on the diagonal and 0s elsewhere.

Example:
Identity matrix I = (egin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix})

Question 9

What is the transpose of a matrix?

Answer 9

Flipping a matrix over its diagonal, switching rows with columns.

Question 10

What is a symmetric matrix?

Answer 10

A matrix that is equal to its transpose.

Question 11

What is a skew-symmetric matrix?

Answer 11

A matrix that is equal to the negation of its transpose.

Question 12

What is the determinant of a matrix?

Answer 12

A scalar value that can be computed from the elements of a square matrix.

Question 13

What is the determinant of a 2x2 matrix?

Answer 13

For matrix (egin{pmatrix} a & b \ c & d end{pmatrix}), the determinant is (ad - bc).

Question 14

What is a cofactor of an element?

Answer 14

The signed minor of an element, used in determinant calculations.

Question 15

What is the adjoint of a matrix?

Answer 15

The transpose of the cofactor matrix.

Question 16

How do you find the inverse of a matrix?

Answer 16

The inverse is found using the formula (A^{-1} = rac{1}{det(A)} ext{adj}(A)).

Question 17

What is a condition for a matrix to be invertible?

Answer 17

The determinant of the matrix must be non-zero.

Question 18

What is Cramer’s Rule?

Answer 18

A method to solve linear systems using determinants.

Question 19

What is the rank of a matrix?

Answer 19

The maximum number of linearly independent row or column vectors.

Question 20

What is row echelon form?

Answer 20

A form where each row has more leading zeros than the previous row.

Question 21

What is Gauss-Jordan elimination?

Answer 21

A method to solve linear systems by transforming the matrix into reduced row echelon form.

Question 22

What is an orthogonal matrix?

Answer 22

A matrix whose rows and columns are orthonormal vectors.

Question 23

What is the trace of a matrix?

Answer 23

The sum of the elements on the main diagonal.

Question 24

Cofactor of an Element

Answer 24

The signed minor of an element, used in determinant calculation.

Question 25

Adjoint of a Matrix

Answer 25

The transpose of the cofactor matrix.

Question 26

Characteristic Equation

Answer 26

det(A - λI) = 0, used to find eigenvalues.

Question 27

Orthogonal Matrices

Answer 27

Matrices whose rows and columns are orthogonal unit vectors.

Question 28

Trace of a Matrix

Answer 28

The sum of the elements on the main diagonal of a square matrix.

Question 29

Rank-Nullity Theorem

Answer 29

Rank(A) + Nullity(A) = Number of columns of A