At least two linear equations in more than two variables.
5x + 3y - 8z = 11
-x + 7y = 10
9x + z = 0
Coefficient Matrix
A matrix consisting of the coefficients of the terms from a system of equations.
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| Coefficient Matrix |
Augmented Matrix
A matrix consisting of the coefficients and constants of the terms from a system of equations.
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| Augmented Matrix |
Side by side comparison
Equal Matrices
Two matrices are equal iff they have the same dimensions and are identical, element by element.
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| Two Equal Matrices |
Zero Matrix aka Identify Matrix for Addition
A matrix whose entries are all zeros.
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| Zero Matrix Zero Matrix Zero Matrix |
Square Matrix
A matrix whose number of rows and columns are equal. A square matrix can also be called a matrix of the nth order (n being the size of the matrix)
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| Square Matrix of 3rd order |
Identity Matrix for Multiplication
A matrix whose entry in the first row and first column is a 1, and entry in the second row and second column is a 1, and whose entry in the third row and third column is a 1, etc. All other entries are 0s.
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| Identity Matrix for Multiplication |
Row-Echelon Form
A matrix is in row echelon form if- Rows consisting entirely of zeros (if any) appear at the bottom of the matrix
- The first entry in a row with nonzero entries is 1, called a leading 1.
- For two consecutive rows with nonzero entries, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.
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| Row-echelon form |
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| Row-echelon form |
Notice in each example, the triangular nature of the nonzero entries.
Reduced Row-Echelon Form
A matrix is in row reduced echelon form if the following conditions are satisfied:
- The first nonzero element in each row (if any) is a "1" (a leading coefficient).
- Each leading coefficient is the only nonzero element in its column.
- All the all-zero rows (if any) are at the bottom of the matrix.
- The leading coefficients form a "stairstep pattern" from northwest to southeast
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