Learn how to easily add and subtract matrix operations. Complete with example questions and step-by-step explanations. Merdeka Curriculum Phase F

A matrix is ​​an arrangement of numbers arranged in rows and columns. Operations on matrices are an important part of what you learn in class XI Phase F of the Independent Curriculum. One of the basic operations on matrices is addition and subtraction.

Study Buddy! Before continuing, make sure you have understood and studied Matrix Concepts and Matrix Types↝ , as well as Matrix Transpose and Similarities↝ yes.

1. Terms for Addition and Subtraction of Matrices

In order for two matrices to be added or subtracted, the two matrices must have the same order. The order of a matrix is ​​the number of rows and columns in the matrix. For example, matrix A with order 2x3 can be added or subtracted from matrix B which also has order 2x3.

2. Matrix Addition

Suppose we have two matrices A and B with the same order, then the addition of matrices A and B can be written as follows:

A+B=[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]+[b11b12⋯b1nb21b22⋯b2n⋮⋮⋱⋮bm1bm2⋯bmn]=[a11+b11a12+b12⋯a1n+b1na21+b21a22+b22⋯a2n+b2n⋮⋮⋱⋮am1+bm1am2+bm2⋯amn+bmn] \begin{align*} A + B &= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} & \cdots & b_{mn} \end{bmatrix} \\&= \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn} \end{bmatrix}\end{align*}

Example:

Given that Matrix A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and Matrix B=[5678] B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}. Determine the result of A+BA+B!

Solution ✍️

A+B=[1234]+[5678]=[1+52+63+74+8]=[681012]\begin{align*}A + B &=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \\&= \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} \\&= \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}\end{align*}

So, the result of A+BA+B is [681012]\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

3. Matrix Subtraction

Matrix subtraction has the same concept as addition. Suppose we have two matrices A and B with the same order, then the reduction of matrices A and B can be written as follows:

A−B=A+(−B)A - B = A + (-B)

Example:

Given that the matrix A=[1234] A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, and the matrix B=[5678] B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} . Determine the outcome of A−BA-B!

Solution ✍️

A−B=[1234]−[5678]=[1−52−63−74−8]=[−4−4−4−4]\begin{align*} A - B &= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} - \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \\ &= \begin{bmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{bmatrix} \\& = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix} \end{align*}

4. Properties of Matrix Addition

  • Commutative: A+B=B+AA + B = B + A
  • Asosiatif: (A+B)+C=A+(B+C)(A + B) + C = A + (B + C)
  • Identity elements: There is a zero matrix (all elements are zero) which, if added to another matrix, will produce itself.

5. Practice Addition and Subtraction of Matrices

So that you better understand matrix addition and subtraction operations, please try the following practice questions

  1. Given the matrix A=[23−14]A = \begin{bmatrix} 2 & 3 \\-1 & 4 \end{bmatrix} and B=[1−250]B = \begin{bmatrix} 1 & -2 \\ 5 & 0 \end{bmatrix} . Calculate A+BA + B.
  2. If the matrix C=[−32140−2]C = \begin{bmatrix}-3 & 2 \\1 & 4 \\0 & -2\end{bmatrix} and D=[5−12310] D = \begin{bmatrix}5 & -1 \\2 & 3 \\1 & 0\end{bmatrix}, determine C - D.
  3. Given the matrix P+Q=[72−15]P + Q = \begin{bmatrix}7 & 2 \\-1 & 5\end{bmatrix} and Q=[3−121]Q = \begin{bmatrix}3 & -1 \\2 & 1\end{bmatrix}. Determine the matrix P.
  4. If the matrix A=[x23y]A = \begin{bmatrix} x & 2 \\ 3 & y \end{bmatrix} and B=[4−125]B = \begin{bmatrix}4 & -1 \\ 2 & 5 \end {bmatrix} and A+B=[7157]A + B = \begin{bmatrix} 7 & 1 \\ 5 & 7 \end{bmatrix}, determine the values ​​of xx and yy.
  5. Given the matrix X=[abcd]X = \begin{bmatrix}a & b \\c & d\end{bmatrix}. If X+X=[681012]X + X = \begin{bmatrix}6 & 8 \\10 & 12\end{bmatrix}, Determine the value of 2a+3d2a + 3d.

Addition and subtraction of matrices are very important basic operations. These two operations can only be performed on matrices that have the same order. By understanding the basic concepts of adding and subtracting matrices, we can perform various more complex matrix operations.