Concept

Addition and Subtraction of Matrices

Matrices can be added and subtracted if they have the same dimensions. The sum and the difference of two $m \times n$ matrices is an $m \times n$ matrix.

Each element of the sum matrix is the sum of the elements of the original two matrices in the same position.

⎣ ⎢ ⎢ ⎡ a_{11} ⋮ a_{m 1} \dots ⋱ \dots a_{1 n} ⋮ a_{m n} ⎦ ⎥ ⎥ ⎤ + ⎣ ⎢ ⎢ ⎡ b_{11} ⋮ b_{m 1} \dots ⋱ \dots b_{1 n} ⋮ b_{m n} ⎦ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎡ a_{11} + b_{11} ⋮ a_{m 1} + b_{m 1} \dots ⋱ \dots a_{1 n} + b_{1 n} ⋮ a_{m n} + b_{m n} ⎦ ⎥ ⎥ ⎤

Each element of the difference matrix is the difference of the elements of the original two matrices in the same position.

⎣ ⎢ ⎢ ⎡ a_{11} ⋮ a_{m 1} \dots ⋱ \dots a_{1 n} ⋮ a_{m n} ⎦ ⎥ ⎥ ⎤ - ⎣ ⎢ ⎢ ⎡ b_{11} ⋮ b_{m 1} \dots ⋱ \dots b_{1 n} ⋮ b_{m n} ⎦ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎡ a_{11} - b_{11} ⋮ a_{m 1} - b_{m 1} \dots ⋱ \dots a_{1 n} - b_{1 n} ⋮ a_{m n} - b_{m n} ⎦ ⎥ ⎥ ⎤

Click on the elements of the sum and difference matrices below to see the illustration of how these operations work on

3 \times 2

matrices.

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