Matrices And Determinants Flashcards
What is a matrix?
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})
What is a row matrix?
A row matrix has only one row.
Example:
Row matrix R = (egin{pmatrix} 1 & 2 & 3 end{pmatrix})
What is a column matrix?
A column matrix has only one column.
Example:
Column matrix C = (egin{pmatrix} 4 \ 5 \ 6 end{pmatrix})
What is a square matrix?
A square matrix has the same number of rows and columns.
Example:
Square matrix S = (egin{pmatrix} 1 & 2 \ 3 & 4 end{pmatrix})
How do you denote a matrix?
A matrix is usually denoted by a capital letter (e.g., A, B, C).
What is matrix addition?
Adding corresponding elements of two matrices of the same dimension.
What is the rule for matrix multiplication?
The number of columns in the first matrix must equal the number of rows in the second matrix.
What is an identity matrix?
A square matrix with 1s on the diagonal and 0s elsewhere.
Example:
Identity matrix I = (egin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix})
What is the transpose of a matrix?
Flipping a matrix over its diagonal, switching rows with columns.
What is a symmetric matrix?
A matrix that is equal to its transpose.
What is a skew-symmetric matrix?
A matrix that is equal to the negation of its transpose.
What is the determinant of a matrix?
A scalar value that can be computed from the elements of a square matrix.
What is the determinant of a 2x2 matrix?
For matrix (egin{pmatrix} a & b \ c & d end{pmatrix}), the determinant is (ad - bc).
What is a cofactor of an element?
The signed minor of an element, used in determinant calculations.
What is the adjoint of a matrix?
The transpose of the cofactor matrix.
How do you find the inverse of a matrix?
The inverse is found using the formula (A^{-1} = rac{1}{det(A)} ext{adj}(A)).
What is a condition for a matrix to be invertible?
The determinant of the matrix must be non-zero.
What is Cramer’s Rule?
A method to solve linear systems using determinants.
What is the rank of a matrix?
The maximum number of linearly independent row or column vectors.
What is row echelon form?
A form where each row has more leading zeros than the previous row.
What is Gauss-Jordan elimination?
A method to solve linear systems by transforming the matrix into reduced row echelon form.
What is an orthogonal matrix?
A matrix whose rows and columns are orthonormal vectors.
What is the trace of a matrix?
The sum of the elements on the main diagonal.
Cofactor of an Element
The signed minor of an element, used in determinant calculation.
Adjoint of a Matrix
The transpose of the cofactor matrix.
Characteristic Equation
det(A - λI) = 0, used to find eigenvalues.
Orthogonal Matrices
Matrices whose rows and columns are orthogonal unit vectors.
Trace of a Matrix
The sum of the elements on the main diagonal of a square matrix.
Rank-Nullity Theorem
Rank(A) + Nullity(A) = Number of columns of A
