Essential Vocabulary
Here is a link to a list of these terms on Quizlet
Adjugate of a matrix (the term Adjoint is also sometimes used)
The transpose of the matrix of cofactors of a matrix - it is part of a formula for the inverse of a matrix.
Cofactor expansion
A method to compute using determinants of minor matrices associated with one row or one column.
Cramer’s rule
A method of solving systems of equations that uses determinants.
Determinant
A function that assigns a scalar output to each square matrix , denoted - it is nonzero if and only if is invertible. Geometrically speaking, the determinant of a linear transformation of a square matrix is the factor by which area (or volume or hypervolume) is scaled by the transformation.
Laplace Expansion Theorem
The determinant of a matrix can be computed using cofactor expansion along ANY row or ANY column.
Properties of determinants
- (a)
- The determinant of a triangular matrix is the product of its diagonal entries.
- (b)
- The determinant of a matrix is equal to the determinant of its transpose.
- (c)
- The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix.
- (d)
- A matrix with a zero row has determinant zero.
- (e)
- Interchanging two rows of a matrix changes the sign of its determinant.
- (f)
- A matrix with two identical rows has determinant zero.
- (g)
- Multiplying a row of a matrix by multiplies the determinant by a factor of .
- (h)
- Multiplying a matrix by multiplies the determinant by a factor of .
- (i)
- Adding a multiple of one row of a matrix to another row does not change the determinant.
- (j)
- A matrix is singular if and only if its determinant is zero.
- (k)
- The determinant of a product is equal to the product of the determinants.
