Matrices and Gauss-Jordan Elimination – Study Notes
Matrix Basics and Notation
A matrix is a rectangular array of numbers with entries denoted by A_{ij}, where i is the row index and j is the column index.
- Notation: A_{ij} is the entry in row i, column j.
- Matrices are denoted by capital letters (e.g., A, B, C).
Size/dimensions
- A ^m x
- A is an m-by-n matrix, written A
- Square matrix: m = n (e.g., 3x3, 4x4)
- Row matrix: 1-by-n (a single row)
- Column matrix (vector): m-by-1
Special matrices
- Zero matrix: all entries are 0
- Identity matrix: I_n is a square matrix with 1s on the main diagonal and 0s elsewhere
Common notations and shapes
- A
- I_n
- A
Matrix operations (key rules)
- You can add or subtract matrices only if they have the same dimensions (A+B and A-B defined when A,B are m-by-n).
- Addition is commutative: A + B = B + A.
- Multiplication is generally not commutative: AB BA in general.
- Scalar multiplication is always allowed: cA is defined for any scalar c.
- Matrix equality: If A = B, then A{ij} = B{ij} for all i,j.
- Matrix product requires compatible inner dimensions: If A is m-by-n and B is n-by-p, then AB is m-by-p and
(AB){ij} = \,\sum{k=1}^{n} A{ik} B{kj} \". - A and B can be real-valued or complex-valued depending on the context.
Examples (illustrative, consistent with standard rules)
- Let A =
\begin{bmatrix} 3 & 4 \ 5 & 6 \end{bmatrix}, B =
\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}. - A + B = \begin{bmatrix} 4 & 6 \ 8 & 10 \end{bmatrix}
- AB = \begin{bmatrix} 31+43 & 32+44 \ 51+63 & 52+64 \end{bmatrix} = \begin{bmatrix} 15 & 22 \ 23 & 34 \end{bmatrix}
- BA = \begin{bmatrix} 13 & 16 \ 29 & 36 \end{bmatrix} (demonstrates non-commutativity in general)
- The identity matrix acts as a multiplicative identity: AIn = InA = A
- Let A =
Meaningful product dimensions (dimension rules)
- If C is 2x3 and D is 3x4, then the product CD is 2x4:
C \in \mathbb{R}^{2\times 3}, \; D \in \mathbb{R}^{3\times 4} \implies CD \in \mathbb{R}^{2\times 4} - If the inner dimensions do not match (e.g., A is m×n and B is p×q with n ≠ p), then AB is not defined (dimension mismatch).
- If C is 2x3 and D is 3x4, then the product CD is 2x4:
Dot product vs matrix product
- The dot product is a specific row-by-row operation associated with a particular pair of vectors, whereas matrix multiplication AB combines rows of A with columns of B using the sum over the inner dimension:
(AB){ij} = \sum{k=1}^{n} A{ik} B{kj} - If A and B have incompatible shapes, the product is undefined.
- The dot product is a specific row-by-row operation associated with a particular pair of vectors, whereas matrix multiplication AB combines rows of A with columns of B using the sum over the inner dimension:
Sizes and compatibility examples (quick checks)
- If A is 2x5 and B is 5x2, then AB is 2x2.
- If A is 2x3 and B is 4x2, AB is undefined (inner dimensions 3 ≠ 4).
Augmented matrices and Gauss-Jordan elimination (Gaussian Elimination)
Goal: Solve systems of linear equations by transforming an augmented matrix to Reduced Row Echelon Form (RREF).
Augmented matrix: concatenate the coefficient matrix A with the right-hand side vector b to get [A|b].
Elementary Row Operations (EROs)
- #1 Row swap: Ri j
- #2 Scalar multiplication: k Ri -> Ri, where k ≠ 0
- #3 Row addition: Ri + t Rj -> R_i, for any scalar t
Reduced Row Echelon Form (RREF)
- Each pivot (the first nonzero in a row) is 1, and all entries above and below each pivot are 0.
- The leftmost nonzero column with a pivot is to the right of the pivot in the previous row, forming a staircase pattern.
- Rows of zeros are at the bottom.
Interpretation of the RREF for solutions
- If the last row is [0 0 0 | b] with b ≠ 0, there is no solution (inconsistent system).
- If there are fewer pivots than unknowns, there are free variables and infinitely many solutions.
- If every variable corresponds to a pivot, there is a unique solution.
How to read off a solution
- From the RREF, write the leading variable (the one with a pivot) in terms of free variables (if any).
Example setup (illustrative): solving a 3-variable system
- System:
- x + y + z = 10000
- 0.05x + 0.08y + 0.09z = 770
- 2x = 7
- Augmented matrix:
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 10000 \
0.05 & 0.08 & 0.09 & 770 \
2 & 0 & 0 & 7
\end{array}\right] - Interpretation and goal: apply EROs to reduce to RREF and read off (x,y,z).
- One possible RREF (example result) would be:
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 3.5 \
0 & 1 & 0 & 12986 \
0 & 0 & 1 & -2989.5 \
\end{array}\right]
which corresponds to the solution $(x,y,z) = (3.5, 12986, -2989.5)$. - Note: The numbers above illustrate the method; actual intermediate steps depend on the exact arithmetic of the row operations.
Special cases to watch for in practice
- If a row reduces to [0 0 0 | b] with b ≠ 0, declare No Solution.
- If there are fewer pivots than unknowns, express some variables as parameters (free variables) to describe the infinite solutions.
- If a row reduces to [0 0 0 | 0], it indicates a dependent equation, contributing to infinitely many solutions.
Example: a system with infinite solutions (dependent equations)
- System:
- x + y - z = 2
- 2x + 2y - 2z = 4
- The second equation is a multiple of the first, so the system has infinitely many solutions.
- Parametric form: Let x = s, y = t (free variables). Then z = x + y - 2 = s + t - 2.
- Solution set: { (s, t, s+t-2) | s,t \in \mathbb{R} }.
Example: a system with a unique solution (3x3)
- System (illustrative):
- x + y + z = 10000
- 0.05x + 0.08y + 0.09z = 770
- 2x = 7
- As shown, this particular system yields a unique triple (x,y,z) depending on arithmetic; one valid solution is (x,y,z) = (3.5, 12986, -2989.5).
Quick checklist for solving with Gauss-Jordan
- Form the augmented matrix [A|b].
- Apply EROs to reach RREF.
- Check for no-solution condition (inconsistent row).
- Determine if the solution is unique or has free variables.
- If needed, express the solution in parametric form for infinite solutions.
Quick practice prompts (from typical coursework and the transcript context)
- For A ∈ R^{m×n} and B ∈ R^{n×p}, show that AB ∈ R^{m×p} and express the entry (AB)_{ij}.
- Given A (2×3) and B (3×4), write the size of AB and provide a concrete example with numbers to compute (AB).
- Form an augmented matrix for a small 2-variable system and reduce to RREF to read off the solution.
- Determine whether a provided system has a unique solution, infinitely many solutions, or no solution based on the RREF.
Connections to foundational principles and real-world relevance
- Matrix operations underpin linear models, systems of equations, and many computational methods used in engineering, economics, computer science, and data analysis.
- Gauss-Jordan elimination is a systematic method for solving linear systems, finding inverses (when applicable), and understanding consistency of equations.
- The concept of inner dimensions in matrix multiplication connects algebraic structure to linear mappings between vector spaces, a foundational idea in linear algebra and applied disciplines.
Notable concepts and terms to remember
- Matrix A ∈ R^{m×n}
- Row operations (EROs): row swaps, scaling, and row additions
- Reduced Row Echelon Form (RREF)
- Augmented matrix [A|b]
- Unique solution vs infinitely many solutions vs no solution
- Inner dimension rule for multiplication
- Identity matrix I_n and zero matrix 0
Quick reference formulas
- Element-wise addition/subtraction: (A\pm B){ij} = A{ij} \pm B_{ij} .
- Scalar multiplication: (cA){ij} = c A{ij}
- Matrix product: (AB){ij} = \sum{k=1}^{n} A{ik} B{kj} \,.
- Size of product: if A \in \mathbb{R}^{m\times n}, \; B \in \mathbb{R}^{n\times p}, then AB \in \mathbb{R}^{m\times p} .
Note on the transcript context
- The notes reflect a typical first-pass curriculum on matrices, including basic notation, types of matrices, operations, and Gaussian elimination, with examples and practice prompts. Some numeric strings in the transcript are garbled, but the core concepts (matrix operations, dimensions, augmented matrices, Gauss-Jordan, and solution types) are preserved and expanded here for clarity and exam preparation.