Find $x$ and $y$, if $2 \begin{bmatrix} 1 & 3 \ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \ 1 & 8 \end{bmatrix}$.

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Accepted Answer

> [!TLDR]
> Matrix addition is element-wise, so we equate corresponding entries after scalar multiplication. Solving the two independent equations gives $x = 3$ and $y = 3$.

Matrix addition works exactly like adding two grids of numbers — you add the entry in row 1, column 1 of the first matrix to the entry in row 1, column 1 of the second, and so on. When a matrix is multiplied by a scalar (like the $2$ here), every entry inside gets multiplied by that scalar first. So the left-hand side becomes a single matrix whose entries are simple expressions in $x$ and $y$. The right-hand side is a known matrix. Since two matrices are equal only when every corresponding entry matches, we get a set of equations — one per position — and solve them.

Let’s go through it step by step.

1. **Multiply the first matrix by the scalar $2$.**  
   $$
   2 \begin{bmatrix} 1 & 3 \ 0 & x \end{bmatrix}
   = \begin{bmatrix} 2 \cdot 1 & 2 \cdot 3 \ 2 \cdot 0 & 2 \cdot x \end{bmatrix}
   = \begin{bmatrix} 2 & 6 \ 0 & 2x \end{bmatrix}
   $$

2. **Add the second matrix to this result.**  
   $$
   \begin{bmatrix} 2 & 6 \ 0 & 2x \end{bmatrix}
   + \begin{bmatrix} y & 0 \ 1 & 2 \end{bmatrix}
   = \begin{bmatrix} 2 + y & 6 + 0 \ 0 + 1 & 2x + 2 \end{bmatrix}
   = \begin{bmatrix} 2 + y & 6 \ 1 & 2x + 2 \end{bmatrix}
   $$

3. **Set this equal to the given right-hand side.**  
   $$
   \begin{bmatrix} 2 + y & 6 \ 1 & 2x + 2 \end{bmatrix}
   = \begin{bmatrix} 5 & 6 \ 1 & 8 \end{bmatrix}
   $$

4. **Equate corresponding entries.**  
   - Top-left: $2 + y = 5$  
   - Top-right: $6 = 6$ (already satisfied, gives no new info)  
   - Bottom-left: $1 = 1$ (also automatically satisfied)  
   - Bottom-right: $2x + 2 = 8$ …

Time	Apples	Bananas
Morning	10	5
Afternoon	8	12

Q.Find $x$ and $y$ , if $2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$ .

Concept understanding — Matrix Addition

Matrix Addition: The Intuition

The Precise Statement

The One Rule You Cannot Break

Properties That Feel Natural

A Quick Check …

Method: Solving for unknowns by equating corresponding entries

Steps

Common Mistakes

Q.Find xxx and yyy, if 2[130x]+[y012]=[5618]2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}2[10​3x​]+[y1​02​]=[51​68​].

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Q.Find $x$ and $y$ , if $2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$ .

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