'+formatMatrix(res);}else if(op==='subtract'){var res=[[0,0,0],[0,0,0],[0,0,0]];for(var i=0;i<3;i++)for(var j=0;j<3;j++)res[i][j]=A[i][j]-B[i][j];resultHtml='Matrix A – B:
'+formatMatrix(res);}else if(op==='multiply'){var res=[[0,0,0],[0,0,0],[0,0,0]];for(var i=0;i<3;i++){for(var j=0;j<3;j++){for(var k=0;k<3;k++){res[i][j]+=A[i][k]*B[k][j];}}}resultHtml='Matrix A × B:
'+formatMatrix(res);}else if(op==='determinant'){var det=A[0][0]*(A[1][1]*A[2][2]-A[1][2]*A[2][1])-A[0][1]*(A[1][0]*A[2][2]-A[1][2]*A[2][0])+A[0][2]*(A[1][0]*A[2][1]-A[1][1]*A[2][0]);if(document.getElementById('fractions').checked)det=det.toFixed(2);resultHtml='Determinant |A| = '+det;}resDiv.innerHTML=resultHtml;}
Calculator Use
This matrix calculator is designed to perform essential linear algebra operations on 3×3 matrices. Whether you are a student solving homework or an engineer working on coordinate transformations, this tool provides instant results for addition, subtraction, multiplication, and determinant calculations.
To use the calculator, simply enter the numerical values for Matrix A and Matrix B (where applicable) and select your desired operation. The results are displayed in a clear grid format, mimicking standard mathematical notation.
- Matrix A & B
- The input grids representing two 3×3 matrices. Each cell corresponds to an element $a_{ij}$ where $i$ is the row and $j$ is the column.
- Operations
- Choose between addition, subtraction, multiplication, or calculating the determinant of the first matrix.
- Rounding
- Use the "Round" checkbox to clean up long decimal results to two decimal places.
How It Works
Matrix operations follow specific rules of linear algebra. Here is how each operation in this matrix calculator is handled:
Matrix Multiplication: $C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$
- Addition: Elements are added component-wise: $A_{ij} + B_{ij}$.
- Subtraction: Elements are subtracted component-wise: $A_{ij} – B_{ij}$.
- Multiplication: The dot product of rows from A and columns from B. Element $C_{11}$ is (Row 1 of A) dot (Column 1 of B).
- Determinant: For a 3×3 matrix, we use the Sarrus rule or cofactor expansion along the first row.
Matrix Calculation Example
Example: Let's add two simple 3×3 matrices where Matrix A and Matrix B both contain 1s in all positions.
Step-by-step solution:
- Identify $a_{11} = 1$ and $b_{11} = 1$.
- Perform addition: $1 + 1 = 2$.
- Repeat for all 9 elements in the 3×3 grid.
- The resulting matrix consists entirely of 2s.
- Final Result = A matrix of [2 2 2; 2 2 2; 2 2 2].
Common Questions
Can this matrix calculator handle larger matrices?
This specific version is optimized for 3×3 matrices, which are the most common in academic problems. For 2×2 matrices, you can leave the third row and column as zeros (though this will affect multiplication and determinants results differently than a true 2×2 tool).
What is a determinant?
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides information about the matrix, such as whether it is invertible (a matrix is invertible if and only if its determinant is non-zero) and the scaling factor of the linear transformation described by the matrix.
Why is matrix multiplication not commutative?
Unlike regular numbers, $A \times B$ does not usually equal $B \times A$. This is because the operation depends on the orientation of rows and columns. Order matters significantly in linear algebra transformations!