Source: MATH 416 Exam 1, Abstract Linear Algebra, UIUC
Tags: systems of linear equations, augmented matrix, row reduction, RREF, reduced row echelon form, Gaussian elimination, matrix multiplication, matrix inverse, transpose, scalar multiplication, free variable, pivot, leading entry
Difficulty: Foundational | Prerequisites: Basic matrix notation, arithmetic with real numbers.
This material covers the two most mechanical (and most tested) skills in a first linear algebra exam: solving systems of linear equations via row reduction, and performing standard matrix arithmetic. If you cannot row-reduce fluently and multiply matrices without errors, every later topic (linear independence, bases, coordinate vectors, linear maps) will trip you up. These operations are the engine room of the course.
Row-reduce the augmented matrix to RREF to read off the solution set of a linear system, using free variables for any non-pivot columns. Matrix multiplication, transposes, scalar multiplication, and inverses follow strict dimension rules, and the 2×2 inverse formula is a favourite exam shortcut.
Augmented matrix
The matrix formed by appending the right-hand-side column vector to the coefficient matrix of a linear system. Written [A | b]. This is the object you row-reduce.
In simple terms, it is the system of equations packed into a single rectangular array so you can do row operations without rewriting variables each time.
Row echelon form (REF)
A matrix where every leading entry (first nonzero entry in a row) is strictly to the right of the leading entry in the row above, and all entries below each leading entry are zero.
Think of it as a staircase pattern going down and to the right.
Reduced row echelon form (RREF)
REF with two extra conditions: every leading entry is 1, and it is the only nonzero entry in its column.
In simple terms, RREF is the "cleanest" form of the matrix, where you can read off variable values (or parametric solutions) directly.
Pivot column
A column of the coefficient matrix that contains a leading 1 in RREF. The corresponding variable is a basic variable (determined by the system).
Free variable
A variable whose column is not a pivot column. It can take any real value, and the solution set is expressed in terms of it.
Matrix multiplication (AB)
Defined when the number of columns of A equals the number of rows of B. The (i, j) entry of AB is the dot product of row i of A with column j of B. The result has the row count of A and the column count of B.
In simple terms, you go across A and down B, multiplying and summing.
Transpose (B^t)
The matrix obtained by swapping rows and columns: the (i, j) entry of B^t is the (j, i) entry of B. If B is m×n, then B^t is n×m.
Inverse of a 2×2 matrix
For A = [[a, b], [c, d]], the inverse is A^{-1} = (1/(ad - bc)) × [[d, -b], [-c, a]], provided ad - bc ≠ 0.
Think of it as: swap the diagonal, negate the off-diagonal, divide by the determinant.
Write the augmented matrix [A | b].
Apply elementary row operations (swap rows, scale a row, add a multiple of one row to another) to reach RREF.
Identify pivot columns and free-variable columns.
If a row reads [0 0 ... 0 | c] with c ≠ 0, the system is inconsistent (no solutions).
Otherwise, express basic variables in terms of free variables. Each free variable introduces one parameter, giving an affine subspace of solutions.
Worked example from the exam. The system
3x₁ + 2x₂ + 3x₃ - 2x₄ = 1 x₁ + x₂ + x₃ = 3 x₁ + 2x₂ + x₃ - x₄ = 2
has augmented matrix:
[ 3 2 3 -2 | 1 ]
[ 1 1 1 0 | 3 ]
[ 1 2 1 -1 | 2 ]
After row reduction, RREF is:
[ 1 0 1 0 | 1 ]
[ 0 1 0 0 | 2 ]
[ 0 0 0 1 | 3 ]
Columns 1, 2, and 4 are pivot columns. Column 3 (x₃) is a free variable. Setting x₃ = t:
x₁ = 1 - t, x₂ = 2, x₃ = t, x₄ = 3
In vector form: (x₁, x₂, x₃, x₄) = (1, 2, 0, 3) + t(-1, 0, 1, 0).
AB is defined only when cols(A) = rows(B).
AB ≠ BA in general; order matters.
Dimensions: if A is m×n and B is n×p, then AB is m×p.
Example. With A (2×2) and B (3×2):
BA is 3×2 times 2×2 = 3×2. AB is not defined (2×2 times 3×2 fails the dimension check). A(B^t) is 2×2 times 2×3 = 2×3.
B + 3B = 4B. Scalar multiplication distributes entry-wise.
Addition requires matrices of the same dimensions.
For A = [[1, 1], [2, 3]]:
det(A) = (1)(3) - (1)(2) = 1. A^{-1} = [[3, -1], [-2, 1]].
Alternative method: augment [A | I] and row-reduce to [I | A^{-1}].
2×2 inverse formula
A^{-1} = (1 / (ad - bc)) × [[d, -b], [-c, a]]
Matrix product entry
(AB){ij} = Σ_k A{ik} B_{kj}
Transpose property
(AB)^t = B^t A^t (note the reversal of order)
Row reduction is the backbone of every numerical solver for linear systems, from engineering simulations to economic input-output models. Matrix multiplication is how graphics engines transform coordinates on screen: every rotation, scaling, and projection is a matrix product.
Students often assume matrix multiplication is commutative (AB = BA). It is not, and in many cases one product is defined while the other is not.
Forgetting to check dimensions before multiplying is a common source of nonsensical answers.
When row-reducing, students sometimes scale a row but forget to apply the same operation to the augmented column, producing a wrong solution set.
A free variable does not mean "no solution." It means infinitely many solutions, parametrised by that variable.
⚠️ Row reduction to RREF and reading off the solution set (including parametric form) appears on nearly every MATH 416 exam.
⚠️ You must show every row operation for full credit. Label each step (e.g. R₂ → R₂ - 3R₁).
⚠️ The 2×2 inverse formula is a fast marks opportunity, but you must verify det ≠ 0 first.
⚠️ Matrix multiplication dimension checks are tested both computationally and conceptually ("is this product defined?").
True or false: If a 3×4 augmented matrix row-reduces to RREF with three pivot columns, the system has a unique solution.
True or false: For any matrices A and B where AB is defined, BA is also defined.
Fill in the blank: If A is 2×3 and B is 3×5, then AB is ×.
True or false: The determinant of [[1, 1], [2, 3]] is 5.
Fill in the blank: A column that does not contain a leading 1 in RREF corresponds to a ___ variable.
Answers: 1. True (3 pivots in a 3-equation system means no free variables). 2. False (dimensions may not match). 3. 2×5. 4. False (it is 1). 5. Free.
Q: A 3×5 augmented matrix [A | b] row-reduces to RREF with pivots in columns 1, 2, and 4. How many free variables are there, and what is the geometric shape of the solution set?
A: The coefficient matrix has 4 columns and 3 pivots, so there is 1 free variable. The solution set is a line (a one-parameter family) in R⁴.
Q: Suppose A is 2×2 with det(A) = 0. What can you say about A^{-1}?
A: A^{-1} does not exist. A square matrix is invertible if and only if its determinant is nonzero.
Q: Given A (2×2) and B (3×2), which of the following are defined: AB, BA, A(B^t), (B^t)A?
A: BA is defined (3×2 times 2×2 = 3×2). A(B^t) is defined (2×2 times 2×3 = 2×3). AB is not defined (columns of A ≠ rows of B). (B^t)A is defined (2×3 times 2×2: not defined, since 3 ≠ 2). Correction: (B^t) is 2×3, A is 2×2, so (B^t)A requires cols(B^t)=3 to equal rows(A)=2, which fails. Not defined.
Q: Write the solution set of the system whose RREF augmented matrix is [[1, 0, 1, 0, 1], [0, 1, 0, 0, 2], [0, 0, 0, 1, 3]].
A: Pivot columns are 1, 2, 4. Column 3 is free. Set x₃ = t. Then x₁ = 1 - t, x₂ = 2, x₃ = t, x₄ = 3. In vector form: (1, 2, 0, 3) + t(-1, 0, 1, 0).
This material connects directly to linear independence (a set of vectors is linearly independent precisely when the homogeneous system with those vectors as columns has only the trivial solution) and to invertibility (a square matrix is invertible if and only if its RREF is the identity). Row reduction also reappears when computing coordinate vectors and change-of-basis matrices.
Gaussian elimination, Gauss-Jordan elimination, echelon form, back substitution, pivot position, basic variable, particular solution, homogeneous system, trivial solution, elementary row operations, row equivalence, matrix product dimensions, determinant, singular matrix, nonsingular matrix, invertible matrix