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Solution of a System

A coordinate point that is a solution to all equations

Solve by Elimination

Method where one variable is removed from the equations so a different one can be solved.

matrix

a rectangular array of elements (or entries) set out by rows and columns

Element (Entry)

A single value value obtained from a matrix, in the form a₍ij₎, where i is the column number and j is the row number.

square matrix

the same number of elements in the rows and columns.

Matrix Addition

If A and B are both matrices of order m × n,

Matrix Subtraction

If A and B are both matrices of order m × n,

matrix multiplication

-can be multiplied if the # of A columns = # of B rows

Inverse of a Square Matrix

For any m × n matrix A = [a₍ij₎], the matrix B such that AB = BA = I₍n₎

Inverse of a 2 × 2 Matrix

If ad - bc ≠ 0, then

Determinant of a Square Matrix

For a n × n matrix A, where n > 2, the sum of the entries in any row or column multiplied by their respective cofactors.

Inverses of n × n Matrices

An n × n matrix has an inverse if and only if set A ≠ 0

Commutative Property

3 + 2 = 2 + 3

Associative Property

(1 + 2) + 3 = 1 + (2 + 3)

Identity Property

the sum of any number and zero is the original number

Inverse Property

The sum of a number and its opposite is 0.

distributive property

Multiplying a sum by a number is the same as multiplying each term by the number and then adding (or subtracting ) the products.

Equivalent Systems

Both systems have the same solution.

Gaussian Elimination

Method of solving systems of equations using the following rules:

Row Echelon Form of a Matrix

A system is in such if:

Partial Fraction Decomposition

For any function f(x) divided by the function d(x), where d(x) has n factors, represented by m,