We will learn how to find the addition of two matrices.

Two matrices A and B are conformable (compatible) for addition if A and B are of the same order.

The sum of A and B is a matrix of the same order and the elements of the matrix A + are obtained by adding the corresponding elements of A and B.

**Example:**

Let A = \(\begin{bmatrix} 12 & 7\\ 3 & -1 \end{bmatrix}\), B = \(\begin{bmatrix} 9 & 3\\ -5 & 4 \end{bmatrix}\), C = \(\begin{bmatrix} 7 & 9 & 5\\ 2 & -3 & 1 \end{bmatrix}\).

(i) A + B can be found because A and B both are of the same order 2 × 2. Adding the corresponding elements,

A + B = \(\begin{bmatrix} 12 + 9 & 7 + 3\\ 3 + (-5) & (-1) + 4 \end{bmatrix}\)

= \(\begin{bmatrix} 21 & 10\\ -2 & 3 \end{bmatrix}\)

(ii) A + C cannot be found because A and C are not of the same order. A is of the order 2 × 2 and C is of the order 2 × 3.

Solved Examples on Addition of Two Matrices

**1.** If A = \(\begin{bmatrix} 1 & 5\\ 7 & 3 \end{bmatrix}\), B = \(\begin{bmatrix} 12 & -1\\ 0 & 9 \end{bmatrix}\), find A + B.

**Solution:**

A + B can be found because A and B both are of the same order 2 × 2.

Now adding the corresponding elements we get,

A + B = \(\begin{bmatrix} 1 & 5\\ 7 & 3 \end{bmatrix}\) + \(\begin{bmatrix} 12 & -1\\ 0 & 9 \end{bmatrix}\)

= \(\begin{bmatrix} 1 + 12 & 5 + (-1)\\ 7 + 0 & 3 + 9 \end{bmatrix}\)

= \(\begin{bmatrix} 13 & 4\\ 7 & 12 \end{bmatrix}\)

**2.** If X = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\), Y = \(\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\), find sum of two matrices X and Y.

**Solution:**

X + Y can be found because X and Y both are of the same order 2 × 2.

Now adding the corresponding elements we get,

X + Y = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\) + \(\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\)

= \(\begin{bmatrix} 1 + 0 & 0 + 1\\ 0 + 1 & 1 + 0 \end{bmatrix}\)

= \(\begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}\)

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