Find X, if $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $2X+Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$.

> [!TLDR] > Matrix addition is element-wise, so we isolate $X$ by subtracting $Y$ from both sides and then dividing by 2. The result is $X = \begin{bmatrix} -1 & -1 \\ -2 & -1 \end{bmatrix}$. The core idea here is that matrix equations work exactly like scalar equations — but only because **matrix addition** is defined component by component. When we have $2X + Y = B$, we can treat each entry independently. There's no matrix multiplication involved, so no worries about order or non-commutativity. Let’s walk through it. 1. **Write the given equation clearly.** We have $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $$2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}.$$ 2. **Isolate the term $2X$.** Subtract $Y$ from both sides. Since matrix subtraction is also element-wise: $$2X = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} - \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}.$$ 3. **Perform the subtraction entry by entry.** - Top-left: $1 - 3 = -2$ - Top-right: $0 - 2 = -2$ - Bottom-left: $-3 - 1 = -4$ - Bottom-right: $2 - 4 = -2$ So $$2X = \begin{bmatrix} -2 & -2 \\ -4 & -2 \end{bmatrix}.$$ 4. **Solve for $X$ by dividing each entry by 2.** Dividing a matrix by a scalar means dividing every element: …

Find X, if Y = 3 2 \1 4 and 2X+Y = 1 0 \-3 2 .

Exercise 3.2 · Q8

Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}

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Concept understanding — Matrix Addition

Matrix Addition: The Intuition

You run a fruit stall and record apples and bananas sold each morning and afternoon in a table:

Time	Apples	Bananas
Morning	10	5
Afternoon	8	12

That's a matrix — a rectangular array of numbers. Your friend's stall has its own table for the same day (morning: 6 apples, 7 bananas; afternoon: 4 apples, 9 bananas). To get the combined sales, you add the numbers in the same position: morning apples with morning apples, afternoon bananas with afternoon bananas, and so on.

That's matrix addition: you add corresponding entries — numbers in the same row and column.

The Precise Statement

(A + B)_{ij} = A_{ij} + B_{ij}

If $A$ and $B$ have the same size (same number of rows and columns), their sum $A + B$ is a matrix of that size where each entry is the sum of the corresponding entries.

Example:

A = \begin{bmatrix} 2 & 5 \\ 1 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 4 \\ 7 & 2 \end{bmatrix}

A + B = \begin{bmatrix} 2+0 & 5+4 \\ 1+7 & -3+2 \end{bmatrix} = \begin{bmatrix} 2 & 9 \\ 8 & -1 \end{bmatrix}

The One Rule You Cannot Break

Watch out

You can only add matrices with the exact same dimensions. A $2 \times 3$ matrix cannot be added to a $3 \times 2$ matrix — the positions don't match.

Properties That Feel Natural

Matrix addition behaves like ordinary number addition, inheriting these from the addition of individual entries:

Commutative: $A + B = B + A$
Associative: $(A + B) + C = A + (B + C)$
Zero matrix: there's a matrix $O$ (all zeros) with $A + O = A$

A Quick Check …

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Matrix addition is element-wise, so we isolate $X$ by subtracting $Y$ from both sides and then dividing by 2. The result is $X = \begin{bmatrix} -1 & -1 \\ -2 & -1 \end{bmatrix}$ .

The core idea here is that matrix equations work exactly like scalar equations — but only because matrix addition is defined component by component. When we have $2X + Y = B$ , we can treat each entry independently. There's no matrix multiplication involved, so no worries about order or non-commutativity.

Let’s walk through it.

Write the given equation clearly. We have $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and

2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}.

Isolate the term $2X$ . Subtract $Y$ from both sides. Since matrix subtraction is also element-wise:

2X = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} - \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}.

Perform the subtraction entry by entry.
- Top-left: $1 - 3 = -2$
- Top-right: $0 - 2 = -2$
- Bottom-left: $-3 - 1 = -4$
- Bottom-right: $2 - 4 = -2$ So

2X = \begin{bmatrix} -2 & -2 \\ -4 & -2 \end{bmatrix}.

Solve for $X$ by dividing each entry by 2. Dividing a matrix by a scalar means dividing every element: …

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Q.Find X, if $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $2X+Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$ .

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: Isolating an unknown matrix in a linear matrix equation

Steps

Common Mistakes

Q.Find X, if Y=[3214]Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}Y=[31​24​] and 2X+Y=[10−32]2X+Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}2X+Y=[1−3​02​].

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Q.Find X, if $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $2X+Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$ .

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