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Matrices - Properties of matrix addition

Grade 12CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Two matrices AA and BB can be added if and only if they are of the same order (same number of rows and columns).

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Commutative Law: Matrix addition is commutative, meaning the order in which two matrices are added does not affect the sum. If A=[aij]A = [a_{ij}] and B=[bij]B = [b_{ij}] are matrices of the same order, then A+B=B+AA + B = B + A.

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Associative Law: For any three matrices A,B,A, B, and CC of the same order, (A+B)+C=A+(B+C)(A + B) + C = A + (B + C). This means the grouping of matrices during addition does not change the result.

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Existence of Additive Identity: For any matrix AA of order m×nm \times n, there exists a zero matrix OO of the same order such that A+O=O+A=AA + O = O + A = A. Here, OO is called the additive identity for matrix addition.

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Existence of Additive Inverse: For any matrix A=[aij]A = [a_{ij}], there exists another matrix −A=[−aij]-A = [-a_{ij}] such that A+(−A)=(−A)+A=OA + (-A) = (-A) + A = O. The matrix −A-A is the additive inverse or negative of AA.

📐Formulae

A+B=B+AA + B = B + A

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

A+O=O+A=AA + O = O + A = A

A+(−A)=OA + (-A) = O

[aij]m×n+[bij]m×n=[aij+bij]m×n[a_{ij}]_{m \times n} + [b_{ij}]_{m \times n} = [a_{ij} + b_{ij}]_{m \times n}

💡Examples

Problem 1:

Given A=[2432]A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} and B=[13−25]B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}, verify the commutative law of addition.

Solution:

First, calculate A+BA + B: A+B=A + B = \begin{bmatrix} 2+1 & 4+3 \ 3+(-2) & 2+5 \end{bmatrix}==\begin{bmatrix} 3 & 7 \ 1 & 7 \end{bmatrix}$$

Next, calculate B+AB + A: B+A=[1+23+4−2+35+2]B + A = \begin{bmatrix} 1+2 & 3+4 \\ -2+3 & 5+2 \end{bmatrix} = \begin{bmatrix} 3 & 7 \ 1 & 7 \end{bmatrix}$$

Since A+B=B+AA + B = B + A, the commutative law is verified.

Explanation:

Matrix addition is performed by adding corresponding elements. Since scalar addition (real numbers) is commutative (a+b=b+aa+b = b+a), matrix addition inherits this property.

Problem 2:

Find the additive inverse of matrix A=[1−204]A = \begin{bmatrix} 1 & -2 \\ 0 & 4 \end{bmatrix} and verify that their sum is the zero matrix.

Solution:

The additive inverse of AA is −A-A, obtained by multiplying every element of AA by −1-1: −A=-A = \begin{bmatrix} -1 & 2 \ 0 & -4 \end{bmatrix}$$

To verify, compute A+(−A)A + (-A): A+(−A)=[1+(−1)−2+20+04+(−4)]A + (-A) = \begin{bmatrix} 1 + (-1) & -2 + 2 \\ 0 + 0 & 4 + (-4) \end{bmatrix} = [0000]\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O$$

Explanation:

The additive inverse −A-A is defined such that the sum of AA and −A-A results in a zero matrix OO of the same order.