Essential Vocabulary

Here is a link to a list of these terms on Quizlet

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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.

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Cofactor expansion

A method to compute \(\det A\) using determinants of minor matrices associated with one row or one column.

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Cramer’s rule

A method of solving systems of equations that uses determinants.

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Determinant

A function that assigns a scalar output to each square matrix \(A\), denoted \(\det A\) - it is nonzero if and only if \(A\) 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.

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Laplace Expansion Theorem

The determinant of a matrix can be computed using cofactor expansion along ANY row or ANY column.

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Properties of determinants

1.
The determinant of a triangular matrix is the product of its diagonal entries.
2.
The determinant of a matrix is equal to the determinant of its transpose.
3.
The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix.
4.
A matrix with a zero row has determinant zero.
5.
Interchanging two rows of a matrix changes the sign of its determinant.
6.
A matrix with two identical rows has determinant zero.
7.
Multiplying a row of a matrix by \(k\) multiplies the determinant by a factor of \(k\).
8.
Multiplying a matrix by \(k\) multiplies the determinant by a factor of \(k^n\).
9.
Adding a multiple of one row of a matrix to another row does not change the determinant.
10.
A matrix is singular if and only if its determinant is zero.
11.
The determinant of a product is equal to the product of the determinants.

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