Linear Systems: Systems Of Equations & Determinants

Front 1

Gauss Solution Matrix

Back 1

When there are 1's on the main diagonal with 0's below (may have #'s or 0's above), the last row provides the solution to the highest number unknown.

Front 2

Gauss-Jordan Solution Matrix

Back 2

There are 1's on the main diagonal with 0's above and below. The last column contains the solution to the corresponding unknowns.

Front 3

Elementary Row Operations

Back 3

(1) Interchange two rows
(2) Multiply a row by a nonzero constant
(3) Add a multiple of one row to another row

Front 4

Row Echelon Form

Back 4

(1) All rows are all 0's or have a leading 1
(2) All 0 rows are grouped at the bottom
(3) Leading 1's of lower rows are further to the right

Front 5

Reduced Row Echelon Form

Back 5

(1) All rows are all 0's or have a leading 1
(2) All 0 rows are grouped at the bottom
(3) Leading 1's of lower rows are further to the right
(4) Each column with a leading 1 has 0's everywhere else.

Front 6

Homogeneous Matrix

Back 6

One in which all the solutions are nonzero, homogenous linear systems are always consistent.

Front 7

Coefficient Matrix

Back 7

Matrix containing the coefficients of the unknowns.

Front 8

Augmented Matrix

Back 8

Coefficient matrix plus an additional column containing constants from the right hand side of the equations.

Front 9

Transpose of a Matrix

Back 9

The original matrix wit its rows and column interchanged.

Front 10

Trace of a Matrix

Back 10

The sum of the entries on the main diagonal of a square matrix.

Front 11

Identity Matrix

Back 11

Square matrix with all 0's except main diagonal contains 1's.

Front 12

Elementary Matrix

Back 12

Square matrix that can be obtained from the identity matrix by performing one single row operation.

Front 13

Diagonal Matrix

Back 13

All entries are 0's except the main diagonal.

Front 14

Upper Triangular Matrix

Back 14

All entries below the main diagonal are 0.

Front 15

Lower Triangular Matrix

Back 15

All entries above the main diagonal are 0.

Front 16

Minor of a Matrix

Back 16

The determinant of the original matrix of the sub-matrix that is left after the row and column of the Minor's position have been removed.

Front 17

Cofactor of a Matrix

Back 17

Equal to the minor times -1 raised to the (row + column) power.

Front 18

Effects of Elementary Row Operations on the Determinant

Back 18

(1) Multiplying a row by a scalar k --> Divide the determinant by k.
(2) Swapping two rows --> Multiply the determinant by -1.
(3) Adding a multiple of one row to another --> No effect.

Front 19

Adjoint of a Matrix

Back 19

The transpose of the Cofactor Matrix. The inverse of the original is equal to the adjoint divided by the determinant.