Choose an operation, set the matrix size, and enter integers, decimals, or fractions in the grid. The result keeps exact values and explains the rule used.
Result
Add corresponding entries
[610812]
Conditions
Matrix A and matrix B both have dimensions 2 by 2.
Each result entry is the sum of entries in the same row and column.
Steps
Check the dimensions Both matrices are 2 by 2, so every entry has a matching position.A,B∈R2×2
Add corresponding entries Add row 1 to row 1 and row 2 to row 2 without changing positions.A+B=[1+53+72+64+8]
Simplify the entries Evaluate each sum to form the result matrix.A+B=[610812]
✓
Independent check
Subtracting B from the result recovers A entry by entry, which independently checks the sum.
Scope
What this matrix operation covers
This matrix calculator handles addition, subtraction, scalar multiplication, transpose, and determinants up to 4 by 4. Addition and subtraction require equal dimensions, while a determinant requires a square matrix.
Add or subtract equal-size matrices
Combine corresponding entries only when the row and column counts match exactly.
Examples: 2 by 2 plus 2 by 2, 2 by 3 minus 2 by 3
Multiply every entry by one scalar
Apply an integer, decimal, or fraction to every position without changing the matrix dimensions.
Examples: -2A, (1/3)A
Transpose a rectangular matrix
Turn rows into columns, so an m by n matrix becomes an n by m matrix.
Examples: 2 by 3 becomes 3 by 2
Find a square-matrix determinant
Calculate an exact determinant for a 1 by 1 through 4 by 4 square matrix.
Examples: 2 by 2 determinant, 3 by 3 cofactor expansion
How to use it
Enter enough information for one clear task
1
Choose the operation first
The form shows only the inputs that operation needs. Addition and subtraction use A and B; scalar multiplication adds a scalar; transpose and determinant use A alone.
2
Set the dimensions
Choose one to four rows and columns. For addition and subtraction, both matrices use the same dimensions. A determinant must be square.
3
Enter one value per cell
Use signed integers, finite decimals, or fractions such as -3/4. Keep the row and column position of each entry unchanged.
4
Read the rule before the result
Confirm the dimensions, inspect the entry-level calculation, and use the independent check to catch an input or arithmetic mistake.
Worked inputs
Examples to try
Use these examples to recognize the method, compare equivalent forms, and check your own work.
Add two 2 by 2 matrices
Add entries that occupy the same row and column.
[1324]+[5768]
Expected result
[610812]
Subtract with negative entries
Subtract each entry of B from the corresponding entry of A.
[42−15]−[1−234]
Expected result
[34−41]
Multiply by a negative scalar
Multiply every entry by negative 2.
−2[12−30]
Expected result
[−2−460]
Transpose a rectangular matrix
Turn each row of A into the matching column of the transpose.
[1−12034]T
Expected result
123−104
Find a 2 by 2 determinant
Use ad minus bc.
det[4276]
Expected result
4(6)−7(2)=10
Find a 3 by 3 determinant
Expand along the first row, whose zero removes one cofactor.
det2−14130021
Expected result
15
Add exact fractions
Add corresponding fractions before simplifying.
[21231−1]+[21−1321]
Expected result
[1110]
Complete example
Find a 3 by 3 determinant without unnecessary work
The first row contains a zero, so expanding across that row leaves only two nonzero cofactors.
A=2−14130021
1
Write the first-row expansion
The signs across the first row are positive, negative, positive.
det(A)=23021−1−1421+0−1430
2
Evaluate the 2 by 2 determinants
Apply ad minus bc inside each remaining minor.
det(A)=2(3⋅1−2⋅0)−((−1)⋅1−2⋅4)
3
Combine the signed terms
The second minor is negative 9, and its cofactor sign is negative.
det(A)=6−(−9)=15
det(A)=15
Verification: A separate elimination-based determinant calculation also gives 15, so the cofactor signs and arithmetic agree.
Avoidable errors
Common mistakes and how to fix them
Adding matrices with different dimensions
Problem: Pair entries until the smaller matrix runs out.
Why it matters: An unpaired entry has no corresponding position, so the sum or difference is undefined.
Better approach: Confirm that both row counts and both column counts match before adding or subtracting.
Moving an entry to a new position
Problem: Reorder entries before adding because the same numbers appear in both matrices.
Why it matters: Matrix position carries meaning. Entry a23 can combine only with b23.
Better approach: Work one fixed row and column position at a time.
Treating scalar multiplication as matrix multiplication
Problem: Use row-by-column products when multiplying A by the number 3.
Why it matters: A scalar is one number, so it multiplies every entry independently.
Better approach: Keep the same dimensions and multiply each aij by the scalar.
Taking a determinant of a rectangular matrix
Problem: Apply the 2 by 2 determinant rule to the first four entries of a 2 by 3 matrix.
Why it matters: A determinant is defined only for a square matrix.
Better approach: Use equal row and column counts before selecting Determinant.
Trust the result for the right reasons
Checks, assumptions, and limits
How results are checked
Exact entry arithmetic is handled by the site's Nerdamer mathematics library.
Addition and subtraction are reversed to recover matrix A.
A transpose is checked by transposing the result a second time.
A determinant is compared with a separate elimination-based numerical pass.
When to stop and revise the input
This release accepts finite real integers, decimals, and simple fractions, not variables or complex entries.
Inputs are limited to 4 by 4 so the steps stay readable and calculations remain responsive.
Matrix-matrix multiplication belongs in the dedicated multiplication calculator.
Inverse matrices, eigenvalues, and row reduction are outside this calculator's current scope.
Common questions
Calculate matrix operations step by step FAQ
When can two matrices be added or subtracted?
They must have exactly the same number of rows and columns. The result has those same dimensions because each output entry combines one matching position from A and B.
Can I enter fractions in the matrix grid?
Yes. Enter a simple fraction such as 3/4 or -2/5. The calculator keeps exact fractional values instead of replacing them with an early decimal approximation.
What happens to dimensions after a transpose?
The row and column counts switch. If A is m by n, then its transpose is n by m because row i becomes column i.
Why is the determinant available only for square matrices?
The determinant is a scalar associated with a square linear transformation. A rectangular matrix has different input and output dimensions, so the square-matrix determinant is not defined.
Where can I multiply two matrices?
Use the matrix multiplication calculator. It gives A and B independent dimensions, checks the inner-dimension rule, and shows every row-by-column dot product.
Sources and curriculum alignment
This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.
