Glossary of Key Matrix Terms
Table of Contents
Matrix
A rectangular array of numbers arranged in rows and columns. For example, a matrix could be 2 rows by 3 columns (written as “2×3 matrix”). Each number in the matrix is called an entry or element. Matrices are used to represent systems of linear equations or transformations in space, among other things.
Element (Entry)
A single number in a matrix, located at a specific row and column. For instance, in a matrix $A$, the element $A_{23}$ refers to the number in the 2nd row, 3rd column of $A$.
Dimension (Size)
The dimensions of a matrix tell how many rows and columns it has. It’s given in the form “(number of rows) × (number of columns).” For example, a 3×4 matrix has 3 rows and 4 columns. Two matrices must have the same dimensions to be added or subtracted. For multiplication, the inner dimensions (columns of the first, rows of the second) must match.
Matrix Addition
Matrix Addition is an operation where two matrices of the same size are added by adding their corresponding entries. The result is a matrix of the same size.
Matrix Subtraction
Matrix Subtraction is an operation where one matrix is subtracted from another of the same size by subtracting each pair of corresponding entries. The result is a matrix of the same size. (It’s effectively adding the first matrix to the negative of the second matrix.)
Matrix Multiplication
Matrix Multiplication is an operation where an $m \times n$ matrix is multiplied by an $n \times p$ matrix to produce an $m \times p$ matrix. Each entry of the product is calculated by taking the dot product of a row of the first matrix with a column of the second matrix. Matrix multiplication has special rules (it’s not commutative, for example). See the Matrix Multiplication section above for a step-by-step explanation.
Dot Product
In the context of matrices, a dot product usually refers to multiplying corresponding entries of a row and a column and summing the results. It’s how we compute each entry of a matrix product. (More generally, a dot product is an operation on two equal-length sequences of numbers, often thought of as vectors, multiplying pairs of numbers and adding them up.)
Augmented Matrix
A matrix that represents a system of linear equations. It includes all the coefficients of the variables and the constants in one matrix. For example, the augmented matrix $\begin{pmatrix}2 & 1 & | & 5 \ 3 & -4 & | & 7\end{pmatrix}$ represents the system $\begin{cases}2x + 1y = 5 \ 3x – 4y = 7\end{cases}$. The vertical bar divides the coefficient matrix and the constants. Augmented matrices are used in methods like Gaussian elimination to solve systems of equations.
Gaussian Elimination
A method for solving systems of linear equations by performing row operations on the augmented matrix of the system. The goal is to get the matrix into an upper triangular form (or echelon form), from which you can easily find the solutions. Our Gaussian Elimination Calculator automates this process, but it’s good to know the term refers to the step-by-step elimination of variables.
Reduced Row Echelon Form (RREF)
A form of a matrix where it has been fully simplified using Gauss-Jordan elimination. In RREF, each leading entry (first non-zero number from the left in a row) is 1, every leading 1 is the only non-zero in its column, and each leading 1 is to the right of the leading 1 in the row above it. A matrix in RREF is very useful – for example, an augmented matrix in RREF makes it easy to read off the solutions to a system of equations. The RREF is a unique final form for a given matrix, no matter how you perform the elimination steps.
Find reduced row-echelon form using the RREF Calculator.
Determinant
A special number that can be calculated from a square matrix (matrices that have the same number of rows and columns). The determinant is denoted often as $\det(A)$ or $|A|$. For a 2×2 matrix $\begin{pmatrix}a & b\ c & d\end{pmatrix}$, the determinant is $ad – bc$. Determinants exist for 3×3, 4×4, etc., with more complex formulas (often computed via expansion or other methods). The determinant tells us certain properties of the matrix – for example, a matrix has an inverse if and only if its determinant is not zero.
Identity Matrix
A special kind of square matrix that acts like “1” does for regular numbers in multiplication. An identity matrix has 1’s on the main diagonal (from top-left to bottom-right) and 0’s everywhere else. For example,
\(
I = \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right)
\)
is the 3×3 identity matrix. If you multiply any matrix $A$ by the identity matrix (of compatible size), you get $A$ back: $I \times A = A$ and $A \times I = A$. The identity matrix is the neutral element for matrix multiplication, just as 0 is the neutral element for addition.
Inverse Matrix
The multiplicative inverse of a square matrix. If $A$ is a square matrix, its inverse (if it exists) is written $A^{-1}$, and it satisfies $A \times A^{-1} = I$ and $A^{-1} \times A = I$, where $I$ is the identity matrix of the same size. Not all matrices have inverses – only those with a non-zero determinant do. Finding an inverse can be done via various methods (such as Gaussian elimination by augmenting $A$ with the identity matrix and reducing to $[I | A^{-1}]$, or using formulas for small matrices). An inverse matrix is like the reciprocal of a number, but for matrices.
Null Space
The null space of a matrix \(A\) is the set of all vectors \(\mathbf{x}\) that satisfy \(A\mathbf{x} = \mathbf{0}\).
It’s a subspace consisting of the solutions to the homogeneous equation. For an \(m \times n\) matrix \(A\), the null space is a subset of \(\mathbb{R}^n\) (the space of column vectors with \(n\) components).
– If \(A\mathbf{x} = \mathbf{0}\) only has the trivial solution \(\mathbf{x} = \mathbf{0}\), the null space is called trivial.
– If there are non-zero solutions, the null space is non-trivial.
The null space is also called the kernel of \(A\). It captures all the linear dependencies among the columns of \(A\): if a vector \(\mathbf{x}\) is in \(\text{Null}(A)\), it means the columns of \(A\) (weighted by the components of \(\mathbf{x}\)) sum to the zero vector.
Kernel
In the context of matrices and linear transformations, the kernel is another word for the null space.
If we view a matrix \(A\) as a linear transformation \(T(\mathbf{x}) = A\mathbf{x}\), then the kernel of \(T\) is
\[
\{ \mathbf{x} \;|\; T(\mathbf{x}) = \mathbf{0} \},
\]
which is exactly the null space of \(A\).
– Kernel vs Null Space: Both terms mean the same thing. “Kernel” is often used in more general abstract settings (like linear transformations between vector spaces), while “null space” is more common when dealing directly with matrices.
– In our calculator, we use null space (kernel) to emphasize this equivalence. If you see “kernel of \(A\),” just remember it is the null space of the matrix \(A\).
Rank
The rank of a matrix \(A\) is the number of linearly independent rows or columns of \(A\).
– These correspond to the maximum number of pivot positions you can obtain after performing Gaussian elimination.
– Rank measures the “information content” of a matrix — how many unique constraints its equations provide.
Examples:
– If \(A\) is a \(4 \times 5\) matrix and has rank 4, that means all 4 rows are independent (and therefore 4 of the 5 columns are also independent).
– If the rank is 3, it means one row (and one column) is redundant, since it can be written as a linear combination of the others.
Importance:
– A high rank (close to the smaller of the number of rows or columns) means most rows/columns are independent.
– A low rank means many rows/columns are redundant.
– In solving \(A\mathbf{x} = \mathbf{b}\), full rank means the system may be solvable for any \(\mathbf{b}\), while less than full rank means \(A\) has inherent limitations.
Rank–Nullity Theorem: In the context of null space, rank helps determine nullity:
\[
\text{rank}(A) + \text{nullity}(A) = \text{number of columns}.
\]
Nullity
Nullity is the term for the dimension of the null space of a matrix.
It is equal to the number of free variables in the homogeneous system \(A\mathbf{x} = \mathbf{0}\).
Formula:
\[
\text{nullity}(A) = \text{number of columns} – \text{rank}(A).
\]
Example: If a matrix \(A\) has 5 columns and rank 3, then
\[
\text{nullity}(A) = 5 – 3 = 2.
\]
This means the null space of \(A\) is 2-dimensional (so there will be 2 basis vectors).
Interpretation:
– Nullity = 0 → the null space has no free directions (only the trivial solution).
– Nullity > 0 → the null space has free variables, giving a count of how many independent “degrees of freedom” the solution space has.
– In other words, nullity tells you how many parameters are needed to describe all solutions.
Note: The calculator output “Nullity (Dimension)” simply reports this number. Checking rank and nullity together reassures you the computation is consistent, thanks to the **rank–nullity theorem.
Pivot
In the context of Gaussian elimination and matrix row reduction, a pivot is a leading entry in a row — typically the first non-zero entry from the left in a row that has been reduced.
– In row echelon form, a pivot is that leading non-zero entry, and each pivot must be to the right of the pivot in the row above it.
– In reduced row echelon form (RREF), pivots are usually equal to 1, and each pivot is the only non-zero entry in its column.
Meaning:
– Pivots correspond to basic variables in a system of equations.
– The number of pivots equals the rank of the matrix.
– Pivot positions show where the matrix has independent information.
Example: If in RREF the first and third columns have pivots (leading 1’s), then \(x_{1}\) and \(x_{3}\) are **basic variables** (determined by the equations), while the other variables are free variables.
When using the calculator’s step-by-step output, you can identify pivot positions by looking at each row’s leading 1. In RREF, all entries below (and above) a pivot are zero.
Why pivots matter: Pivots are critical in forming the solution: they allow you to express pivot (basic) variables in terms of free variables.
Basis
A basis in linear algebra is a set of vectors that are linearly independent and span a given space.
– In the context of the null space, a null space basis is a set of vectors \(\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{p}\}\) such that:
1. Each \(\mathbf{v}_{i}\) is in \(\text{Null}(A)\) (i.e. \(A\mathbf{v}_{i} = \mathbf{0}\)),
2. The vectors are independent (no \(\mathbf{v}_{i}\) can be formed as a combination of the others), and
3. Together, they span the entire null space (any solution \(\mathbf{x}\) with \(A\mathbf{x} = \mathbf{0}\) can be written as a combination of the \(\mathbf{v}_{i}\)’s).
The number of vectors in the basis equals the nullity of \(A\).
Examples:
– If \(\text{Null}(A)\) is a line through the origin in \(\mathbb{R}^3\), a basis could be one non-zero vector pointing along that line.
– If \(\text{Null}(A)\) is a plane through the origin, a basis might be two independent vectors that span the plane.
Notes:
– A basis is not unique — many different sets of vectors can form a basis as long as they meet the conditions.
– Typically, the calculator returns one convenient basis (often chosen by setting each free variable to 1 in turn).
– Having a basis is useful because it gives a parametric description of the null space: instead of describing all solutions, you only provide a few basis vectors, and any solution is obtained by combining them.