If $\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}$, then the value of $x + y$ is (A) $x = 3,\ y = 1$ (B) $x = 2,\ y = 3$ (C) $x = 2,\ y = 4$ (D) $x = 3,\ y = 3$

> [!TLDR] > Two matrices are equal if and only if their corresponding entries are equal. Equating the four entries gives a system of equations in $x$ and $y$; solving yields $x=2$, $y=3$, so $x+y=5$. We are given the matrix equation: $$ \begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix} $$ The core idea is simple: **matrix equality** means every entry in the same position must be equal. This turns one matrix equation into four ordinary equations. --- 1. **Equate the (1,1) entries** (top-left): $$ 2x + y = 7 $$ 2. **Equate the (1,2) entries** (top-right): $$ 4x = 7y - 13 $$ 3. **Equate the (2,1) entries** (bottom-left): $$ 5x - 7 = y $$ 4. **Equate the (2,2) entries** (bottom-right): $$ 4x = x + 6 $$ Now we have four equations, but they must be consistent — we only need enough to find $x$ and $y$. > [!TIP] > Always start with the simplest equation. Here, equation (4) is just $4x = x+6$, which gives $x$ immediately. From equation (4): $$ 4x = x + 6 \quad\Rightarrow\quad 3x = 6 \quad\Rightarrow\quad x = 2 $$ Now substitute $x=2$ into equation (1): $$ 2(2) + y = 7 \quad\Rightarrow\quad 4 + y = 7 \quad\Rightarrow\quad y = 3 $$ Check with equation (3) to be safe: $$ 5(2) - 7 = 10 - 7 = 3 = y \quad\checkmark $$ …

If 2x+y 4x \5x-7 4x = 7 7y-13 y x+6 , then the value of x + y is (A) x = 3,y = 1 (B) x = 2,y = 3 (C) x = 2,y =

NCERT Exemplar · Q53

\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}

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Concept understanding — Matrix Notation

Matrix Notation: A First Look

You run a fruit stall selling apples, bananas, and oranges, and you track sales across Monday, Tuesday, and Wednesday. You could write three separate lists:

Monday: 10 apples, 5 bananas, 8 oranges
Tuesday: 7 apples, 12 bananas, 3 oranges
Wednesday: 9 apples, 6 bananas, 11 oranges

That works but is messy. A matrix is a cleaner way to arrange the same information — a rectangular grid of numbers organised into rows and columns.

The Intuition: A Table of Numbers

Think of a matrix as a spreadsheet:

Rows represent one category (a day of the week)
Columns represent another (a type of fruit)

For the stall:

\begin{bmatrix} 10 & 5 & 8 \\ 7 & 12 & 3 \\ 9 & 6 & 11 \end{bmatrix}

The first row is Monday's sales; the second column holds all banana sales across the three days: 5, 12, 6.

Note

A matrix is not a number — it's a collection of numbers in a specific shape. You can't say "this matrix equals 5" any more than "this table equals 5."

The Precise Statement

A matrix is a rectangular array of numbers arranged in $m$ rows and $n$ columns — an $m \times n$ matrix (read "m by n"). The numbers inside are its entries or elements. To name a specific entry we use two subscripts: row first, then column.

A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}

Here $a_{ij}$ is the entry in row $i$ , column $j$ ; for example $a_{23}$ is second row, third column.

A = [a_{ij}]_{m \times n}

Shorthand for "the matrix $A$ whose entry in row $i$ , column $j$ is $a_{ij}$ , with $m$ rows and $n$ columns."

Key Vocabulary

Term	Meaning	Example
Order (size)	rows × columns	$3 \times 3$ for the fruit matrix
Square matrix	same number of rows and columns	$2 \times 2$ , $3 \times 3$
Row matrix	only one row	$[1 \quad 2 \quad 3]$ (a $1 \times 3$ matrix)
Column matrix	only one column	$\begin{bmatrix}4 \\ 5 \\ 6\end{bmatrix}$ (a $3 \times 1$ matrix)
Zero matrix	all entries are 0	$\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$

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Two matrices are equal if and only if their corresponding entries are equal. Equating the four entries gives a system of equations in $x$ and $y$ ; solving yields $x=2$ , $y=3$ , so $x+y=5$ .

We are given the matrix equation:

\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}

The core idea is simple: matrix equality means every entry in the same position must be equal. This turns one matrix equation into four ordinary equations.

Equate the (1,1) entries (top-left):

2x + y = 7

Equate the (1,2) entries (top-right):

4x = 7y - 13

Equate the (2,1) entries (bottom-left):

5x - 7 = y

Equate the (2,2) entries (bottom-right):

4x = x + 6

Now we have four equations, but they must be consistent — we only need enough to find $x$ and $y$ .

Tip

Always start with the simplest equation. Here, equation (4) is just $4x = x+6$ , which gives $x$ immediately.

From equation (4):

4x = x + 6 \quad\Rightarrow\quad 3x = 6 \quad\Rightarrow\quad x = 2

Now substitute $x=2$ into equation (1):

2(2) + y = 7 \quad\Rightarrow\quad 4 + y = 7 \quad\Rightarrow\quad y = 3

Check with equation (3) to be safe:

$5(2) - 7 = 10 - 7 = 3 = y \quad\checkmark$ …

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Q.If $\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}$ , then the value of $x + y$ is
(A) $x = 3,\ y = 1$
(B) $x = 2,\ y = 3$
(C) $x = 2,\ y = 4$
(D) $x = 3,\ y = 3$

Concept understanding — Matrix Notation

Matrix Notation: A First Look

The Intuition: A Table of Numbers

The Precise Statement

Key Vocabulary

Method: Finding unknowns from equality of two matrices

Steps

Common Mistakes

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