8.1 Vocab

Cramer's Rule

Using determinants to write the solution of a system of linear equations.

Collinear Points

Triangular Matrix

Points are on the same line

A square matrix with elements sij = 0 for j < i is termed upper triangular matrix. In other words, a square matrix is upper triangular if all its entries below the main diagonal are zero.

Cofactor

Cryptogram

The determinant obtained by deleting the row and column of a given element of a matrix or determinant. The cofactor is preceded by a + or – sign depending whether the element is in a + or – position.

A message written according to a secret code.

Entry

A real number within a matrix.

Each a in the picture is an entry.

Matrix

A rectangular array of numbers, symbols, or expressions

Rows

Minor

A horizontal line in a matrix, represented by the variable m.

Elementary Row Operations:

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

Row switching

A row within the matrix can be switched with another row.

Row multiplication

Each element in a row can be multiplied by a non-zero constant.

Row addition

A row can be replaced by the sum of that row and a multiple of another row.

If E is an elementary matrix, as described below, to apply the elementary row operation to a matrix A, one multiplies the elementary matrix on the left, E⋅A. The elementary matrix for any row operation is obtained by executing the operation on the identity matrix

A minor M_(ij) is the reduced determinant of a determinant expansion that is formed by omitting the ith row and jth column of a matrix A.

Columns

A Vertical line in a matrix, represented by the variable n

Cryptography

The art of writing or solving codes.

Main Diagonal

Inverse

The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A^(-1) such that

AA^(-1)=I,

Augmented Matrix

Coefficient Matrix

A matrix derived from a system linear equations, where each row is from one equation.

First column: x terms

Second column: y terms.

Third column: z terms.

Final column: Constant.

A matrix derived from coefficients of the system of equations, not including constant terms.

Similar to augmented matrix, but without constant column.

Works Cited

Elementary Row Operations

Shifts in an augmented matrix that are legal:

1. Interchange two rows

2. Multiply a row by a nonzero constant

3. Add a multiple of a row to another row.

Determinant

A real number that can associate with a square matrix.

8.1 Vocab (Period 5)

by Joe, Jacky, Alec S., Mancy, and Fifi.

Row Equivalent

Two matrices are row equivalent if one can be reached through a series of elementary row operations upon the other.

http://thejuniverse.org/PUBLIC/LinearAlgebra/LOLA/matCalc/index.html

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"DrDelMath." College Algebra Chapter 6 Summary. N.p., n.d. Web. 29 Jan. 2014.

"DrDelMath." College Algebra Chapter 6 Summary. N.p., n.d. Web. 29 Jan. 2014.

"Diagonal." -- from Wolfram MathWorld. N.p., n.d. Web. 29 Jan. 2014.

"List of Matrices." Wikipedia. Wikimedia Foundation, 21 Jan. 2014. Web. 29 Jan. 2014. <http://en.wikipedia.org/wiki/List_of_matrices>.

http://mathonweb.com/help/mat11.gif

http://www.mathsisfun.com/algebra/images/matrix-gauss-jordan2.gif

"Mathwords: Gaussian Elimination." Mathwords: Gaussian Elimination. N.p., n.d. Web. 29 Jan. 2014. <http://www.mathwords.com/g/gaussian_elimination.htm>.

"The Substitution Method." How to Solve Systems of Linear Equations by Substitution , Examples, Pictures, Practice| Math Warehouse. N.p., n.d. Web. 29 Jan. 2014. <http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/solve-by-substitution.php>.

"Matrix (mathematics)." Wikipedia. Wikimedia Foundation, 23 Jan. 2014. Web. 29 Jan. 2014. <http://en.wikipedia.org/wiki/Matrix_%28mathematics%29>.

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"Determinant of a Matrix." Tutorvista.com. N.p., n.d. Web. 29 Jan. 2014. <http://www.tutorvista.com/content/math/determinant-of-a-matrix/>.

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http://www.smart-kit.com/s1835/cryptogram-2/

http://www.mathopenref.com/collinear.html

http://www.easilearn.com/us/ViewArticle2.aspx?TextSize=Large&ContentId=19706

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http://enchantedmind.com/images/creative_cryptography_graphic_b.gif

x2

[4 10]

[2 5]

x 1/2

Matrix Operations

Addition, division, and multiplication

Row Echelon Form

A matrix is in row-echelon form when the following conditions are met:

1. If there is a row of all zeros, then it is at the bottom of the matrix.

2. The first non-zero element of any row is a one. That element is called the leading one.

3. The leading one of any row is to the right of the leading one of the previous row.

4. All elements below or above a leading one are zero

Infinite Solution

Any number would satisfy the variables in a scenario like this; there are infinitely many solutions

Leading 1

Gaussian Elimination

A 1 in a row that is the first nonzero number in its row.

A strategy of solving systems of equations by making the system into a matrix and reducing it to row-echelon form.