MATRICES

Matrix: is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Example of a 2 x 3 matrix ("2x3" read as "two by three")


Some definition/terms about this lessons:
Matrix: is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Real Matrix: is a matrix whose elements consist of entirely of real numbers
Augmented Matrix: is a matrix obtain by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on the each of the given matrices.
Coefficient Matrix: refers to a matrix consisting of the coefficients of the variables in a set of linear equations.
Elementary row operation:

• Interchange two of the equations in the system.
• Multiply one of the equations through by a number other than zero.
• Add a multiple of one equation to another.

Square Matrix: Have the same number of rows and columns.
Rows: The horizontal line of numbers.
Columns: The vertical line of numbers.
Row Echelon form: a matrix is in echelon form if it has the shape resulting of a Gaussian elimination.

-All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix).
-The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it (some texts add the condition that the leading coefficient must be 1.
-All entries in a column below a leading entry are zeroes (implied by the first two criteria).
Example:




Reduced-Row Echelon Form:  A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:
-It is in row echelon form.
-Every leading coefficient is 1 and is the only nonzero entry in its column.

        Example: