Finite Math Test 1 Key Terms

What is the General Form of the equation of a line?

AX + BY + C = 0

What is the Point-Slope Form of the equation of a line?

y - y₁ = m(x - x₁)

What is the Slope-Intercept Form of the equation of a line?

y = mx + b

What are the forms of the equation for 2 lines in a 2×2 linear system?

1) ax + by = h
2) cx + dy = k

What is an alternative term for the point that satisfies both lines in a 2×2 linear system of equations?

The point of equilibrium

What are the 3 possible solutions for a 2×2 linear system of equations?

1) One unique solution (intersecting lines)
2) No solution (parallel lines)
3) Infinitely many solutions (coincidental lines)

Finish the three legal operations of the method of elimination:
1) Switch .
2) Multiply an equation by .
3) Multiply an equation by and add it to .

1) Switch any two equations.
2) Multiply an equation by a nonzero number.
3) Multiply an equation by a nonzero number and add it to another equation.

What are the two methods of solving a system of equations without using matrices? Explain the first steps of each.

1) Substitution: Focus on one equation and solve it for “x“, then plug this solution into the second equation for “x.“
2) Elimination: Multiply one equation by a constant and add it to the second equation to cancel out one of the unknowns.

What is an augmented matrix?

An augmented matrix joins the coefficient matrix with the constant matrix.

Finish the 4 characteristics of a Row-Reducted Form of a matrix:
1) Each row consisting entirely of lies all rows having entries.
2) The first entry in each (nonzero) row is (leading ).
3) In any two successive (nonzero) , the leading in the lower row lies to the of the leading in the upper row.
4) If a column in the coefficient matrix contains a leading , then the other entries in that column are .

1) Each row consisting entirely of zeroes lies below all rows having nonzero entries.
2) The first nonzero entry in each (nonzero) row is 1 leading 1).
3) In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row.
4) If a column in the coefficient matrix contains a leading 1, then the other entries in that column are zeroes.

Finish the 3 characteristics of the Gauss-Jordan Method:
1) Switch any two .
2) Multiply any by a nonzero .
3) Multiply a by a nonzero and it to another .

1) Switch any two rows.
2) Multiply any row by a nonzero number .
3) Multiply a row by a nonzero number and add it to another row.

An overdetermined system vs. an underdetermined system

Overdetermined: More equations than unknowns
Underdetermined: Fewer equations than unknowns

The possible solutions for an overdetermined system versus the possible solutions for an undetermined system

Overdetermined: No solution, exactly one solution, infinitely many solutions
Underdetermined: Infinitely many solutions or no solution

What is a square system?

The number of unknowns is equal to the number of equations

The “m x n“ of a matrix designates its number of and .

Rows x columns

If matrix A is an “m x n“ matrix, the matrix “n x m“ matrix of A is called the of A, or .

Transpose, A^T

If A is an m x n matrix and B is an n x p matrix, then AB is and is an x matrix.

If A is an m x n matrix and B is an n x p matrix, then AB is defined and is an m x p matrix.

I_n is called the of size .

I_n is called the identity matrix of size n.

A^-1 is the of A.

A^-1 is the inverse of A.

How do you find the inverse for matrix A if it exists? Why does this work?

1) Set matrix A and the identity matrix side-by-side.
Solve A to see if you can get the identity matrix.
If you can, the inverse of A exists and is the matrix to the right of A.
2) A^-1A=I and A^-1I=B, so B=A^-1.

If A^-1 exists, A is called .
If A^-1 does not exist, then A is called .

If A^-1 exists, A is called nonsingular.
If A^-1 does not exist, then A is called singular.

What is the difference between the transpose (A^T) and inverse (A^-1) of a matrix?

Transpose: Flips the matrix on a diagonal by swapping the rows and columns of the original matrix for the columns and rows of the second matrix.
Inverse: Specific to square matrices and, when multiplied by the original matrix, produces the identity matrix. (A^-1A = I)

Proof of the process to find the inverse of a matrix

AX=B
(A^-1)AX=B(A^-1)
Because A^-1A=I,
Then IX=B(A^-1)
I=1
Tf X=A^-1B