Matrix theory

The Matrices are two-dimensional or rectangular row and column arrangements that are represented as follows:

Matrices

Where every aij it is an element of the matrix.

By knowing what a matrix represents, we can talk about some of its forms, for example:

Type of matrices

Zero matrix

It is an array where all its elements are zero:

Matrix types - zero matrix

Square matrix

An array is square when the number of rows equals its number of columns (n = m).

Sum of matrices

If we want to add, we must be clear that the only way for such an operation to occur is that both matrices have the same number of rows and columns (n = m), otherwise they cannot be added together.

The sum of matrices is equal to the sum of the elements corresponding to each matrix, that is:

matrix sum

Example:

We have the following matrices:

\ large A _ {\ left (2 \ times 2 \ right)} = \ begin {vmatrix} 2 & 1 \\ 2 & 4 \\ \ end {vmatrix}

\ large B _ {\ left (2 \ times 2 \ right)} = \ begin {vmatrix} 3 & 4 \\ 7 & 0 \\ \ end {vmatrix}

\ large D _ {\ left (n \ times m \ right)} = \ begin {vmatrix} 1 & 4 \\ 5 & 3 \\ 9 & 2 \ end {vmatrix}

If we wanted to perform the addition operation with these, we realize that only matrices A and B can be added, since matrix D does not have the same number of rows as A and B. Then we will add only A and B:

\ large A _ {\ left (2 \ times 2 \ right)} + B _ {\ left (2 \ times 2 \ right)} = \ begin {vmatrix} 2 + 3 & 1 + 4 \\ 2 + 7 & 4 + 0 \ \ \ end {vmatrix} = \ begin {vmatrix} 5 & 5 \\ 9 & 4 \\ \ end {vmatrix}

Multiplication of matrices by scalars

Now let's see how a matrix is multiplied by a scalar (real numbers). We say we have a matrix A(n × m) multiplied by α:

scalar matrix multiplication

Then each element of matrix A is multiplied by α:

\ large A _ {\ left (n \ times p \ right)} = \ begin {vmatrix} a_ {i1} & a_ {i2} & a_ {ip} \ end {vmatrix}

\ large B _ {\ left (p \ times m \ right)} = \ begin {vmatrix} b_ {1j} \\ b_ {2j} \\ b_ {pj} \ end {vmatrix}

\ large \ begin {vmatrix} a_ {i1} & b_ {i2} & b_ {ip} \ end {vmatrix} \ cdot \ begin {vmatrix} b_ {1j} \\ b_ {2j} \\ b_ {pj} \ end { vmatrix} = a_ {i1} \ cdot b_ {1j} + a_ {i2} \ cdot b_ {2j} + a_ {ip} \ cdot b_ {jp}

Matrix multiplication

For this operation to be possible, the first matrix must have the number of columns equal to the number of rows of the second matrix, that is, A(n × p)· B(p × m) the result of this multiplication must be a matrix that has the n rows of A and the m columns of B. Then:

A(n × p)· B(p × m) = C(n × m)

Example:

\ large A _ {\ left (2 \ times 2 \ right)} = \ begin {vmatrix} 1 & 4 \\ 3 & 4 \\ \ end {vmatrix}

\ large B _ {\ left (2 \ times 2 \ right)} = \ begin {vmatrix} 2 & 3 \\ 1 & 0 \\ \ end {vmatrix}

These two matrices can be multiplied, because the number of columns in A is equal to the number of rows in B. Let's multiply:

\ large A \ cdot B = \ begin {vmatrix} 1 & 2 \\ 3 & 4 \\ \ end {vmatrix} \ cdot \ begin {vmatrix} 2 & 3 \\ 1 & 0 \\ \ end {vmatrix} = \ begin {vmatrix } \ left (1 \ cdot 2 \ right) + \ left (2 \ cdot 1 \ right) & \ left (1 \ cdot 3 \ right) + \ left (2 \ cdot 0 \ right) \\ \ left (3 \ cdot 2 \ right) + \ left (4 \ cdot 1 \ right) & \ left (3 \ cdot 3 \ right) + \ left (4 \ cdot 0 \ right) \\ \ end {vmatrix}

\ large A \ cdot B = C = \ begin {vmatrix} 4 & 3 \\ 10 & 9 \\ \ end {vmatrix}

It is important to know that this operation is not commutative, that is to say that:

\ large A \ cdot B \ neq B \ cdot A

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