What Are The Properties Of Matrices?
The properties of matrices are the algebraic laws the four matrix operations obey: addition, scalar multiplication, multiplication, and transpose. Each operation has its own short list of rules, and together they tell you what you are allowed to do when rearranging a matrix expression.
Variable glossary. $A, B, C$ denote matrices; $O$ is the zero matrix (all entries 0); $I$ is the identity matrix; $k, l$ are scalars (ordinary numbers). The order of a matrix is its rows-by-columns count, and most properties require the orders to be compatible.
The fastest way to hold all of this is by operation, so the rest of the article takes them one at a time.
What Are The Properties Of Matrix Addition?
Matrix addition is the most well-behaved operation — it inherits every rule from ordinary number addition, as long as the matrices share the same order.
Commutative: $A + B = B + A$. Order of addition does not matter.
Associative: $(A + B) + C = A + (B + C)$. Grouping does not matter.
Additive identity: $A + O = O + A = A$. Adding the zero matrix changes nothing.
Additive inverse: $A + (-A) = O$. Every matrix has a negative that cancels it.
Closure: the sum of two $m \times n$ matrices is again an $m \times n$ matrix.
All of these hold because addition is done entry by entry, and each entry is just an ordinary number obeying the ordinary rules.
What Are The Properties Of Scalar Multiplication?
Multiplying a matrix by a scalar $k$ scales every entry by $k$, and the operation distributes cleanly over both matrices and scalars.
Distributive over matrix addition: $k(A + B) = kA + kB$.
Distributive over scalar addition: $(k + l)A = kA + lA$.
Associative with scalars: $(kl)A = k(lA) = l(kA)$.
Sign rule: $(-k)A = -(kA) = k(-A)$, so $(-1)A = -A$.
The order $m \times n$ is preserved: scaling never changes a matrix's dimensions.
What Are The Properties Of Matrix Multiplication?
This is where matrices break from numbers, and the break is the most important single fact in the article.
Matrix multiplication is not commutative: in general $AB \neq BA$. Swapping the factors usually gives a different answer, and sometimes one product is defined while the other is not. The order in which you multiply matrices carries meaning.
The properties that do hold:
Associative: $(AB)C = A(BC)$. Grouping is safe.
Left and right distributive: $A(B + C) = AB + AC$ and $(B + C)A = BA + CA$. (Notice the second keeps $A$ on the right — because order matters, you cannot collapse these into one.)
Multiplicative identity: $AI = IA = A$, with $I$ the identity matrix of the matching order.
Compatibility: the product $AB$ exists only when the number of columns of $A$ equals the number of rows of $B$.
There is no general "multiplicative inverse" property: only some matrices — the invertible ones — have an inverse $A^{-1}$ with $A A^{-1} = I$.
What Are The Properties Of The Transpose?
The transpose interacts with the other operations through four rules, and one of them mirrors the non-commutativity above.
Double transpose: $(A^T)^T = A$.
Over addition: $(A + B)^T = A^T + B^T$.
Over scalars: $(kA)^T = k A^T$.
Over multiplication (order reverses): $(AB)^T = B^T A^T$.
That last rule reverses the factor order for exactly the same reason multiplication is non-commutative — the orders have to stay compatible. A related identity, $\det(A^T) = \det(A)$, ties the transpose back to the determinant. The bigger picture of how operations and determinants fit together sits in matrices and determinants.
Examples Of Properties Of Matrices
Example 1
Verify $A + B = B + A$ for $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$.
Add in each order, entry by entry:
$A + B = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$
$B + A = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$
Final answer: Both give $\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$, so addition is commutative.
Example 2
Show $k(A + B) = kA + kB$ for $k = 2$, $A = \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & 1 \ 0 & 4 \end{bmatrix}$.
Left side: add first, then scale.
$A + B = \begin{bmatrix} 4 & 1 \ 2 & 5 \end{bmatrix}$
$2(A + B) = \begin{bmatrix} 8 & 2 \ 4 & 10 \end{bmatrix}$
Right side: scale each, then add.
$2A = \begin{bmatrix} 2 & 0 \ 4 & 2 \end{bmatrix}$
$2B = \begin{bmatrix} 6 & 2 \ 0 & 8 \end{bmatrix}$
$2A + 2B = \begin{bmatrix} 8 & 2 \ 4 & 10 \end{bmatrix}$
Final answer: Both sides equal $\begin{bmatrix} 8 & 2 \ 4 & 10 \end{bmatrix}$, confirming the distributive property.
Example 3
Check whether $AB = BA$ for $A = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 \ 3 & 1 \end{bmatrix}$.
The natural assumption, carried over from numbers, is that the product is the same either way.
Wrong assumption. Expect $AB = BA$ because "multiplication commutes."
Test it. Compute both.
$AB = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \ 3 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 2 \ 3 & 1 \end{bmatrix}$
$BA = \begin{bmatrix} 1 & 0 \ 3 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 \ 3 & 7 \end{bmatrix}$
Why the assumption fails. The two results differ, so $AB \neq BA$. Matrix multiplication depends on order.
Final answer: $AB = \begin{bmatrix} 7 & 2 \ 3 & 1 \end{bmatrix}$ and $BA = \begin{bmatrix} 1 & 2 \ 3 & 7 \end{bmatrix}$; they are not equal.
Example 4
Verify the distributive property $A(B + C) = AB + AC$ for $A = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}$, $C = \begin{bmatrix} 0 & 2 \ 1 & 0 \end{bmatrix}$.
Left side: add inside, then multiply.
$B + C = \begin{bmatrix} 1 & 3 \ 1 & 1 \end{bmatrix}$
$A(B + C) = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix}\begin{bmatrix} 1 & 3 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 6 \ 4 & 6 \end{bmatrix}$
Right side: multiply each, then add.
$AB = \begin{bmatrix} 2 & 2 \ 1 & 4 \end{bmatrix}$
$AC = \begin{bmatrix} 0 & 4 \ 3 & 2 \end{bmatrix}$
$AB + AC = \begin{bmatrix} 2 & 6 \ 4 & 6 \end{bmatrix}$
Final answer: Both sides equal $\begin{bmatrix} 2 & 6 \ 4 & 6 \end{bmatrix}$, so the distributive law holds.
Example 5
Confirm $(AB)^T = B^T A^T$ for $A = \begin{bmatrix} 1 & 2 \ 3 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 & 1 \ 4 & 2 \end{bmatrix}$.
Compute $AB$, then transpose:
$AB = \begin{bmatrix} 8 & 5 \ 0 & 3 \end{bmatrix}$
$(AB)^T = \begin{bmatrix} 8 & 0 \ 5 & 3 \end{bmatrix}$
Now $B^T A^T$, reversing the order:
$B^T = \begin{bmatrix} 0 & 4 \ 1 & 2 \end{bmatrix}$, $A^T = \begin{bmatrix} 1 & 3 \ 2 & 0 \end{bmatrix}$
$B^T A^T = \begin{bmatrix} 0 & 4 \ 1 & 2 \end{bmatrix}\begin{bmatrix} 1 & 3 \ 2 & 0 \end{bmatrix} = \begin{bmatrix} 8 & 0 \ 5 & 3 \end{bmatrix}$
Final answer: Both equal $\begin{bmatrix} 8 & 0 \ 5 & 3 \end{bmatrix}$, so $(AB)^T = B^T A^T$.
Example 6
Show $A + O = A$ and $AI = A$ for $A = \begin{bmatrix} 7 & -2 \ 5 & 9 \end{bmatrix}$.
Adding the zero matrix leaves $A$ unchanged:
$A + O = \begin{bmatrix} 7 & -2 \ 5 & 9 \end{bmatrix} + \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 7 & -2 \ 5 & 9 \end{bmatrix}$
Multiplying by the identity also leaves $A$ unchanged:
$AI = \begin{bmatrix} 7 & -2 \ 5 & 9 \end{bmatrix}\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & -2 \ 5 & 9 \end{bmatrix}$
Final answer: $A + O = A$ and $AI = A$; $O$ is the additive identity, $I$ the multiplicative identity.
Why The Properties Of Matrices Matter: "Knowing which rules survive the jump"
The properties of matrices exist to mark the boundary between the algebra of numbers, which students already know, and the algebra of matrices, which mostly looks the same but is not. The whole point is to flag the one place the familiar rules break, so you do not carry a false assumption into a problem.
Where this boundary does real work:
Solving systems. Methods that solve $AX = B$ rely on associativity and the identity property — but they must respect non-commutativity, since you cannot freely move $A$ past $X$. The order of operations is the method.
Computer graphics and robotics. A rotation followed by a translation is not the same as a translation followed by a rotation. That is the non-commutative property made physical: the order you apply transformations changes where the object ends up.
Transposing expressions. Simplifying $(ABC)^T$ correctly to $C^T B^T A^T$ — not $A^T B^T C^T$ — keeps least-squares and covariance computations valid.
Students who assume every number-rule transfers will write $AB = BA$ and get plausible-looking nonsense. The first instinct on a matrix product is to commute it the way you would commute $3 \times 5$ — and that single carried-over habit is the most common matrix error there is. At Bhanzu, trainers teach the properties as a "which rules survive?" checklist, so the non-commutative exception is the thing a student remembers rather than the thing they forget.
Where Students Slip On Matrix Properties (And How To Fix It)
Mistake 1: Assuming matrix multiplication commutes
Where it slips in: Any product where it feels natural to swap factors.
Don't do this: Replace $AB$ with $BA$, or "simplify" $ABA^{-1}$ to $B$.
The correct way: Keep the order. $AB \neq BA$ in general, so factors cannot be reordered. The first instinct is to treat matrices like numbers and swap freely — the habit comes straight from years of commuting ordinary products, and it is exactly what breaks. When in doubt, leave the order alone.
Mistake 2: Forgetting to reverse the order in a transposed product
Where it slips in: Transposing $AB$ or a longer product.
Don't do this: Write $(AB)^T = A^T B^T$.
The correct way: Reverse the factors: $(AB)^T = B^T A^T$. The memorizer who learned "transpose each piece" drops the reversal. Tie it to the order rule — the dimensions only stay compatible when the factors flip.
Mistake 3: Adding matrices of different orders
Where it slips in: Reaching for a sum before checking dimensions.
Don't do this: Add a $2 \times 3$ matrix to a $3 \times 2$ matrix entry by entry.
The correct way: Addition needs identical orders; otherwise the sum is undefined. The rusher who jumps straight to adding entries skips the order check. Confirm both matrices are $m \times n$ before adding a single entry.
Key Takeaways
The properties of matrices are the rules for four operations: addition, scalar multiplication, multiplication, and transpose.
Addition is commutative, associative, and has the zero matrix as its identity; scalar multiplication distributes over both matrices and scalars.
Matrix multiplication is associative, distributive, and has the identity $I$ — but it is not commutative: $AB \neq BA$ in general.
The transpose rules are $(A^T)^T = A$, $(A + B)^T = A^T + B^T$, $(kA)^T = kA^T$, and $(AB)^T = B^T A^T$.
The most common error is assuming the number-rule of commutativity carries over to matrix multiplication.
A Practical Next Step
Work through these to test which rules you trust, then return to the multiplication section if non-commutativity keeps catching you:
Pick any two $2 \times 2$ matrices and confirm $AB \neq BA$ by computing both.
Verify $(A + B)^T = A^T + B^T$ for a pair of your choice.
Check that $k(A + B) = kA + kB$ for $k = 3$.
Treat "is this product order-sensitive?" as your first question on every matrix expression.
To master matrix operations with a teacher who flags exactly which rules carry over, work with an algebra tutor, get help with algebra, or join math classes online. Want a live Bhanzu trainer to walk through more matrix-property problems? Book a free demo class.
Read More
Solve matrices — where associativity and the identity property power the solving methods.
Diagonal matrix — a structured family on which several properties simplify.
Adjoint of a matrix — built from cofactors and the transpose, used to invert a matrix.
Symmetric matrix — the family defined by the transpose property $A = A^T$.
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