Matrices operations calculator

Operations on two matrices

Matrix A

Scalar multiply:

Determinant:
Rank:
Trace:

Matrix B

Scalar multiply:

Determinant:
Rank:
Trace:

Result Matrix C

Scalar multiply:

Determinant:
Rank:
Trace:

NOTES

Calculator notes:

Operations on matrices A anb B are denoted by highlighting the operation button [+] [−] or [X] by bold green.

Any operation done on matrix C will disconnect the operation button to indicate that matrix C is no longer a result of an operation on matrices A and B.

Calculations like: 2A^-1 + 3B^T can be performed easily, it is possible to extend the calculation capabilities by the copy operation.

The arithmetic buttons are enabled to indicate the allowed operations according to the sizes of the matrices.

Example: Find the value of the expression A² − 4A − 5 I.

if A =

1	2
3	4

Solution:

Step 1:

Enter matrix A values.

Step 2:

Copy matrix A to matrix B.

Step 3:

Press on matrix A square button A².

Step 4:

Enter the value 4 into scalar multiply of matrix B.

Step 5:

Press on the minus button.

Step 6:

Copy the result of matrix C into matrix A.

Step 7:

At matrix B update the scalar multiply to 1.

Step 8:

At matrix B press the unit I matrix button.

Step 9:

At matrix B enter the value 5 in the scalar multiply.

Step 10:

Because the minus button is already illuminated by the last operation, C contains the final answer.

-2	2
3	1

Matrices Types

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Matrices overview

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The notation of a matrix of size (m ✕n) is defined as A(m ✕ n) = A(row, column)

A convenient shorthand which offers considerable advantage when working with system of linear equations is by using the matrix notation. Consider the set of linear equations of the form:

In matrix notation these equations may be represented as:

or AX = C

The terms of the matrix can be represented as:

Distributive law

Left side A(B + C) = AB + AC

Right side (A + B)C = AC + BC

A(m ✕ n) B and C (n ✕ p)

A and B (m ✕ n) C(n ✕ p)

Associative law

Addition (A + B) + C = A + (B + C)

Multiplication (AB)C = A(BC)

(m ✕ n)

(n ✕ n)

Scalar multiplication

(kA)B = A(kB) = k(AB)

A(m ✕ n) B(n ✕ p)
k any number

Commutative law

Addition A + B = B + A

Multiplication not commutative

A and B (m ✕ n)

Because A∙B ≠ B∙A

Other algebraic laws

(k, v are constants)

0 + A = A	k(A + B)= kA + kB
1 · A = A	(k + v)A = kA + vA
0 · A = 0	k(vA) = (kv)A
A + ( A) = 0	kA = 0 → k = 0 or A = 0
(  1)A =  A

Matrices powers

(c is constant)

Matrices addition and multiplication

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Matrices addition:
A and B are of the same size m × n

Scalar multiplication

Matrices multiplication A (m × n) ∙ B (n × p) = C (m × p)

Example:

Determinants

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Determinants - symbol: det A or |A|

The result of the determinant of a matrix (n ⨯ n) is a real number.

Size 2 matrix
Size 3 matrix


General form to evaluate determinant values:

In this formula M_in is the determinant of the submatrix of A obtained by deleting its ith row and nth column. The determinant M_in is called the minor of the element a_in and his size is (n-1) ⨯ (n-1).

Cofactors of matrix A_ij

It is convenient to consolidate the quantity (-1)^i+j and the minor M_ij . We define the cofactor A_ij of the element a_ij in determinant A as: A_ij = (-1)^i+jM_ij .

Determinant’s properties:

Example: Evaluate the determinant:	det A = 1(12-(-1)(-1))-(-2)(22-(-1)(-2))+3(2(-1)-1(-2))
	det A = 1(2-1)+2(4-2)+3(-2+2) = 5

Transposed matrix A^T

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Transposed matrix A^T	Interchange of terms across the main diagonal

Transposed matrices properties:

Example: Find the transposed of the matrix
	Note: The transposed size of an m ⨯ n matrix is n ⨯ m.

Inverse matrix A^-1

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Inverse matrix A^-1 = B

The matrix A is inversible if there is a matrix B so that: AB = BA = I
then the matrix B is the inversed matrix of A.
Matrix I is the unit matrix. Thus the solution of A X = B can be written in the form X = A^-1 B (where A is an n x n matrix and X and B are n x 1 matrices).

Inversed matrices properties:

Example: Find the inverse of matrix A	1. Add the unit matrix at the right:

	2. Multiply first row by -2 and add it to the second row then multiply first row by -4 and add it to the third row to obtain:

	3. Add second and third rows to obtain:

	4. Subtract third row from second row:

	5. Finally multiply third row by 2 and add it to the first row and multiply third row by -1 to get the unit matrix:

	And the inverse of A is:

Rank of a matrix A

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Rank of a matrix A

A square matrix is said to be non-singular, if its determinant is not zero. The rank of an m ⨯ n matrix is the largest integer r for which a non-singular
r ⨯ r submatrix exists.
If A and B be an n ⨯ n matrices then: rank(A + B) ≤ rank A + rank B

Example: Find the rank of matrix A.	1. Multiply first row by 2 and add it to the second row. 2. Multiply first row by -3 and add it to the third row. 3. Subtract fourth row from the first row to get:

	4. Add second row to the third row. 5. Subtract fourth row from second row to obtain:
	The result matrix is:
	And the r ⨯ r non zero matrix is:

	And the rank of matrix A is 3.

Scaling Matrices

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Enlarging or shrinking a vector can be done by multiplying the vector by the diagonal matrix of the form:

If a = b = c > 1	then the vector is enlarged equally in all directions.
If a = b = c < 1	then the vector is shrinking equally in all directions.
If a ≠ b ≠ c	then the vector is scaled in different sizes in the x, y and z directions.