UNIT 8 • MATRICES & LINEAR ALGEBRA

Matrix Calculator
Add, Multiply, Transpose & Powers

A matrix calculator that shows its working. Add or subtract matrices of the same size; multiply when the columns of A match the rows of B. Each entry of a product is row i of A times column j of B.

THE RULESSIZE MATTERS

(A + B)_{ij} = a_{ij} + b_{ij} \qquad (AB)_{ij} = \sum_{k} a_{ik}\, b_{kj} \qquad (A^{T})_{ij} = a_{ji}

1. Check the SizesAdd/subtract: same size. Multiply: an

m \times n

times an

n \times p

gives

m \times p

2. Row × ColumnFor each entry of AB, multiply row i of A by column j of B, entry by entry, and add.

3. Order MattersIn general

AB \neq BA

, so keep the order the question gives.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find $AB$ for $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ .

STEP 01 • CHECK THE SIZES

A is 2×2 and B is 2×2: the inner numbers match (2 = 2), so AB exists and is 2×2.

STEP 02 • ROW 1 OF A × EACH COLUMN OF B

c_{11} = (1)(5) + (2)(7) = 19, \qquad c_{12} = (1)(6) + (2)(8) = 22

STEP 03 • ROW 2 OF A × EACH COLUMN OF B

c_{21} = (3)(5) + (4)(7) = 43, \qquad c_{22} = (3)(6) + (4)(8) = 50

FINAL CONCLUSION

AB = [[19, 22], [43, 50]]

AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Try B × A in the calculator: the answer is different, because matrix multiplication depends on the order.

LABORATORY

Interactive Matrix Calculator

LABORATORY

Matrix Operations Engine

Operation

Rows:Columns:

A =

Rows:Columns:

B =

Common Questions

How do you multiply two matrices?

Entry (i, j) of

AB

is row i of A times column j of B: multiply the matching entries and add. The number of columns of A must equal the number of rows of B.

Is AB the same as BA?

Usually not. Matrix multiplication is not commutative, and BA may not even exist when AB does. With the example on this page,

BA = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}

, which is different from AB.

What is the transpose of a matrix?

The matrix you get by turning rows into columns:

(A^{T})_{ij} = a_{ji}

. A 2×3 matrix becomes 3×2.

When can two matrices be added?

Only when they have the same size. You then add the entries in the same position.

PreviousRank & Echelon Form

Unit 8•38 of 47

NextArea & Perimeter

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 8 • MATRICES & LINEAR ALGEBRA

Matrix Calculator
Add, Multiply, Transpose & Powers

THE RULESSIZE MATTERS

(A + B)_{ij} = a_{ij} + b_{ij} \qquad (AB)_{ij} = \sum_{k} a_{ik}\, b_{kj} \qquad (A^{T})_{ij} = a_{ji}

1. Check the SizesAdd/subtract: same size. Multiply: an

m \times n

times an

n \times p

gives

m \times p

2. Row × ColumnFor each entry of AB, multiply row i of A by column j of B, entry by entry, and add.

3. Order MattersIn general

AB \neq BA

, so keep the order the question gives.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find $AB$ for $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ .

STEP 01 • CHECK THE SIZES

A is 2×2 and B is 2×2: the inner numbers match (2 = 2), so AB exists and is 2×2.

STEP 02 • ROW 1 OF A × EACH COLUMN OF B

c_{11} = (1)(5) + (2)(7) = 19, \qquad c_{12} = (1)(6) + (2)(8) = 22

STEP 03 • ROW 2 OF A × EACH COLUMN OF B

c_{21} = (3)(5) + (4)(7) = 43, \qquad c_{22} = (3)(6) + (4)(8) = 50

FINAL CONCLUSION

AB = [[19, 22], [43, 50]]

AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Try B × A in the calculator: the answer is different, because matrix multiplication depends on the order.

LABORATORY

Interactive Matrix Calculator

LABORATORY

Matrix Operations Engine

Operation

Rows:Columns:

A =

Rows:Columns:

B =

Common Questions

How do you multiply two matrices?

Entry (i, j) of

AB

is row i of A times column j of B: multiply the matching entries and add. The number of columns of A must equal the number of rows of B.

Is AB the same as BA?

Usually not. Matrix multiplication is not commutative, and BA may not even exist when AB does. With the example on this page,

BA = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}

, which is different from AB.

What is the transpose of a matrix?

The matrix you get by turning rows into columns:

(A^{T})_{ij} = a_{ji}

. A 2×3 matrix becomes 3×2.

When can two matrices be added?

Only when they have the same size. You then add the entries in the same position.

PreviousRank & Echelon Form

Unit 8•38 of 47

NextArea & Perimeter

Matrix Calculator Add, Multiply, Transpose & Powers

Step-by-Step Worked Example

AB = [[19, 22], [43, 50]]

Interactive Matrix Calculator

Matrix Operations Engine

Common Questions

NETZ Engine

Matrix Calculator Add, Multiply, Transpose & Powers

Step-by-Step Worked Example

AB = [[19, 22], [43, 50]]

Interactive Matrix Calculator

Matrix Operations Engine

Common Questions

Matrix Calculator
Add, Multiply, Transpose & Powers

Matrix Calculator
Add, Multiply, Transpose & Powers