Operations on Matrices

Addition of Matrices

If A and B are two matrices of the same order, then A and B are said to be Conformable or Compatible for addition. The sum of A and B is denoted by A+B and it is obtained by adding corresponding elements of A & B.

The matrix A+B will be of the same order as each of the matrices A and B is

If A= \(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}\) and B= \(\begin{bmatrix} b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\end{bmatrix}\)

then A+B =\(\begin{bmatrix} a_{11}+b_{11}&a_{12}+b_{12}&a_{13}+b_{13}\\a_{21}+_b{21}&a_{22}+b_{22}&a_{23}+b_{23}\end{bmatrix}\)

Foe example, if A= \(\begin{bmatrix} 1&2&3\\4&-3&7\end{bmatrix}\) and B= \(\begin{bmatrix} -3&4&7\\8&9&-3\end{bmatrix}\)

Then,

A+B =\(\begin{bmatrix} 1&2&3\\4&-3&7\end{bmatrix}\) + \(\begin{bmatrix} -3&4&7\\8&9&-3\end{bmatrix}\)

= \(\begin{bmatrix} 1-3&2+4&3+7\\4+8&-3+9&7-3\end{bmatrix}\)

= \(\begin{bmatrix} -2&6&10\\12&6&4\end{bmatrix}\)

Subtraction of Matrices

If A and B are two matrices of the same order, then they are said to be conformable for subtraction. The difference of the matrix B from A is denoted by A-B and it is obtained by subtracting the elements of B from the corresponding elements of A.

The order of the matrix A-B is same as the order of A or B.

If A = \(\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22} &a_{23}\end{bmatrix}\) and B = \(\begin{bmatrix}b_{11}&b_{12}&b_{13}
\\b_{21}&b_{22}&b_{23}\end{bmatrix}\)

then A-B = \(\begin{bmatrix} a_{11}-b_{11}&a_{12}-b_{12}&a_{13}-b_{13}\\a_{21}-b_{21}&a_{22}-b_{22}&a_{23}-b_{23}\end{bmatrix}\)

Foe example, if A =\(\begin{bmatrix} 1&2&3\\4&-3&7\end{bmatrix}\) and B= \(\begin{bmatrix} -3&4&7\\8&9&-3\end{bmatrix}\)

Then

A-B=\(\begin{bmatrix} 1&2&3\\4&-3&7\end{bmatrix}\) and B= \(\begin{bmatrix} -3&4&7\\8&9&-3\end{bmatrix}\)

=\(\begin{bmatrix} 1+3&2-1&3-7\\4-8&-3-9&7+3\end{bmatrix}\)

=\(\begin{bmatrix} 4&-2&-4\\-4&-12&10\end{bmatrix}\)

B-A =\(\begin{bmatrix} -3&4&7\\8&9&-3\end{bmatrix}\) -\(\begin{bmatrix} 1&2&3\\4&-3&7\end{bmatrix}\)

=\(\begin{bmatrix} -3-1&4-2&7-3\\8-4&9+3&-3-7\end{bmatrix}\)

=\(\begin{bmatrix} -4&2&4\\4&-12&-10\end{bmatrix}\)

Multiplication of a matrix by a scalar (real number)

If A is any matrix and K is any constant or a scalar, then the matrix obtained by multiplying each element of A by K is denoted by KA and it is called the scalar multiple of A by K. 

Algebraic properties of Matrix Addition

Addition of matrices satisfy the following properties:

Closure property

If A and B are two matrices of the same order, then their A+B is also a matrix of the same order as that of A or B.

If A=\(\begin{bmatrix} 2&4\\3&5\end{bmatrix}\) and B=\(\begin{bmatrix} 6&5\\2&1\end{bmatrix}\)

Now,

A+B = \(\begin{bmatrix} 2&4\\3&5\end{bmatrix}\) + \(\begin{bmatrix} 6&5\\2&1\end{bmatrix}\)

= \(\begin{bmatrix} 8&9\\5&6\end{bmatrix}\) which is again is again a2x2 matrix. Hence the closer property is satisfied.

Commutative property

If A and B are two matrices of the same order, then A+B=B+A.

Let, A = \(\begin{bmatrix} 2&4\\6&8\end{bmatrix}\) + \(\begin{bmatrix} 7&-2\\8&4\end{bmatrix}\)

Then A+B =\(\begin{bmatrix} 2&4\\6&8\end{bmatrix}\) +\(\begin{bmatrix} 7&-2\\8&4\end{bmatrix}\)

=\(\begin{bmatrix} 9&2\\14&12\end{bmatrix}\)

B+A =\(\begin{bmatrix} 7&-2\\8&4\end{bmatrix}\) +\(\begin{bmatrix} 2&4\\6&8\end{bmatrix}\)

=\(\begin{bmatrix}9&2\\14&12\end{bmatrix}\)

∴ A+B=B+A

Hence the commutative property is satisfied.

Associative property

If A, B and C three matrices of the same order, then (A+B) + C = A +(B+C)

Existence of identity element

If A is any matrix, then three exists a null matrix 0 of the same order such that A+0 = 0+A=A.

Existence of additive inverse

If A is a matrix of any order, then there exists another matrix -A of same order such that A+(-A) =(-A) +A = 0, then additive identity.

If A and B are the matrices of the same order and K is a scalar, then

K(A+B) = KA+KB.

If A is a matrix and C,K are any two scalars, then (C+K) A=CA+KA

If C,K are any two scalars and A is a matrix, then C(KA) = (CK)A.

Solving Matrix Equations

To solve the matrix equation A + X =B where A and B are two given matrices of the same order and X is unknown, we proceed in a manner similar to numbers.

Here, A + X = B

Adding the matrix (-A)to both sides, we get

(-A)+ A+ X = (-A) + B

or, (-A+A) +X = B-A

or, 0 + X = B-A

or, X = B-A, which is the required solution.

Transpose of matrix

Let A be a matrix. Then the new matrix obtained by interchanging the corresponding rows and column of A is called the transpose of A. It is denoted by A' or A^t.

If A = \(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}\)

Then A^t= \(\begin{bmatrix} a_{11}&a_{12}\\a_{13}&a_{21}\\a_{22}&a_{23}\end{bmatrix}\)

Here, the order of A' is 2 x 3 and that of A^t is 3 x 2. Hence, if the order of matrix A is mxn, then the order of A^t will be nxm. If A' is a square, matrix of order n, then A^t is also a square matrix of order n. If A' is a row matrix, then A^t is a column matrix.

Properties of transpose

1. The transpose of the transpose of a matrix is itself. .i.e. (a') = A.
2. The transpose of the sum of two matrices is the sum of their transposes. i.e (A+)' = A' +B'
3. If A is any matrix and K is any number, then (KA)' = KA'.