Step 1: Establishing the Equality of Matrices

Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are said to be equal if and only if:

• They are of the same order.
• Their corresponding elements are equal for all possible values of $i$ and $j$ ($a_{ij} = b_{ij}$).

[Definition of Equality of Matrices]

Step 2: Verification of Matrix Order

The given matrices are:

$A = \begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix}$ (Order: $2 \times 2$)

$B = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$ (Order: $2 \times 2$)

Since both matrices have an order of $2 \times 2$, the equality is well-defined. [Per the condition of equality for matrices of same dimensions].

Step 3: Equating Corresponding Elements

By equating the elements at each position $(i, j)$ from matrix $A$ to matrix $B$:

Visual Representation of Matrix Equality

Step 4: Final Derivation

From the equations derived in Step 3, we directly extract the values:

• $x = 1$
• $y = 4$
• $z = 3$

Final Solution: x = 1, y = 4, z = 3

Step 1: Matrix Definition and Structural Setup

A $3 \times 4$ matrix $A$ consists of $3$ rows and $4$ columns. The general form of the matrix is represented as:

$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{bmatrix}$

Each element $a_{ij}$ is defined by the rule $a_{ij} = 2i - j$, where $i$ denotes the row index ($i \in \{1, 2, 3\}$) and $j$ denotes the column index ($j \in \{1, 2, 3, 4\}$).

Step 2: Calculating Individual Elements

We evaluate each element systematically by substituting the indices into the given formula $a_{ij} = 2i - j$:

Step 3: Matrix Assembly

Substituting the calculated values into the general matrix structure:

$A = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 3 & 2 & 1 & 0 \\ 5 & 4 & 3 & 2 \end{bmatrix}$

Visualization of Index Mapping

Final Solution: The constructed matrix is $A = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 3 & 2 & 1 & 0 \\ 5 & 4 & 3 & 2 \end{bmatrix}$

Step 1: Identifying the Matrix Structure

The given matrix $A$ is presented as an array of elements organized into horizontal rows and vertical columns:

By inspection, we define the dimensions of the matrix as follows:

• Rows ($m$): These are the horizontal arrangements of elements. Counting them:
  • Row 1: $[2, 5, 19, -7]$
  • Row 2: $[35, -2, \frac{5}{2}, 12]$
  • Row 3: $[\sqrt{3}, 1, -5, 17]$
  Thus, the number of rows $m = 3$.
• Columns ($n$): These are the vertical arrangements of elements. Counting them:
  • Column 1: $[2, 35, \sqrt{3}]^T$
  • Column 2: $[5, -2, 1]^T$
  • Column 3: $[19, \frac{5}{2}, -5]^T$
  • Column 4: $[-7, 12, 17]^T$
  Thus, the number of columns $n = 4$.

Step 2: Defining the Order of a Matrix

[Per the definition of Matrix Algebra], the order of a matrix is denoted as $m \times n$, where $m$ represents the number of rows and $n$ represents the number of columns. This notation identifies the dimensions of the rectangular array.

Step 3: Calculating the Final Order

Substituting the identified values into the $m \times n$ format:

Final Solution: The order of the matrix $A$ is $3 \times 4$.

Step 1: Identifying Matrix Specifications

The given problem specifies a matrix of order $m \times n$, where $m = 3$ and $n = 3$. The total number of entries (elements) in the matrix is determined by the product of the number of rows and the number of columns [Definition of Matrix Order].

Total number of entries ($N$) = $m \times n = 3 \times 3 = 9$.

Step 2: Defining the Sample Space of Entries

For each individual position $(i, j)$ in the $3 \times 3$ matrix, the allowed values are restricted to the set $\{0, 1\}$.

• The number of choices for each entry is $k = 2$.

Step 3: Calculating Total Permutations

Since each of the $N$ positions in the matrix is independent and can be filled by any of the $k$ choices, the total number of possible matrices is given by the Fundamental Counting Principle [Permutation and Combination Axiom].

Total matrices = $k \times k \times \dots \times k$ ($N$ times)

Total matrices = $k^N$

Step 4: Algebraic Computation

Substitute the identified values into the derived formula:

$k = 2$

$N = 9$

Total matrices = $2^9$

Final Solution: The number of all possible $3 \times 3$ matrices with entries 0 or 1 is $512$.

Step 1: Establishing the Condition for Equality

Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are said to be equal if and only if they are of the same order and their corresponding elements are equal [Definition of Equality of Matrices]. Both given matrices are of order $2 \times 2$. Thus, we equate corresponding elements:

$\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix} = \begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$

Step 2: Formulating Algebraic Equations

Equating the elements at specific positions $(i, j)$:

• For element $a_{11} = b_{11}$: $3x + 7 = 0$ (Equation 1)
• For element $a_{12} = b_{12}$: $5 = y - 2$ (Equation 2)
• For element $a_{21} = b_{21}$: $y + 1 = 8$ (Equation 3)
• For element $a_{22} = b_{22}$: $2 - 3x = 4$ (Equation 4)

Step 3: Solving for Unknowns

Solving for $x$:

From (Equation 1):

From (Equation 4):

Solving for $y$:

From (Equation 2):

From (Equation 3):

Step 4: Analytical Verification

For the matrices to be equal, the value of $x$ must satisfy both (Equation 1) and (Equation 4) simultaneously. As derived, (Equation 1) yields $x = -\frac{7}{3}$ while (Equation 4) yields $x = -\frac{2}{3}$. Since these values are inconsistent ($-\frac{7}{3} \neq -\frac{2}{3}$), there exists no single value of $x$ that satisfies the equality for both elements $a_{11}$ and $a_{22}$ concurrently.

Since the requirements for $x$ are contradictory, the matrix equality cannot be satisfied for any pair $(x, y)$.

Final Solution: The values of $x$ and $y$ for which the matrices are equal cannot be determined; therefore, the correct option is "Not possible to find".

Step 1: Theoretical Foundation of Matrix Construction

A matrix $A$ of order $m \times n$ is a rectangular array of numbers arranged in $m$ rows and $n$ columns. For a $2 \times 2$ matrix, the general form is defined as:

where $i$ denotes the row index and $j$ denotes the column index, with $i, j \in \{1, 2\}$. [Definition of a Matrix, NCERT Mathematics Textbook for Class XII].

Step 2: Systematic Calculation of Individual Elements

Given the governing formula $a_{ij} = \frac{(i+j)^2}{2}$, we calculate each element systematically:

• For $a_{11}$ ($i=1, j=1$):
  $a_{11} = \frac{(1+1)^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2$
• For $a_{12}$ ($i=1, j=2$):
  $a_{12} = \frac{(1+2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5$
• For $a_{21}$ ($i=2, j=1$):
  $a_{21} = \frac{(2+1)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5$
• For $a_{22}$ ($i=2, j=2$):
  $a_{22} = \frac{(2+2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8$

Step 3: Matrix Assembly

Substituting the computed values back into the general $2 \times 2$ matrix structure:

Visual Representation

Final Solution: The matrix $A$ is given by $\begin{bmatrix} 2 & 4.5 \\ 4.5 & 8 \end{bmatrix}$.

Step 1: Theoretical Foundation

An $m \times n$ matrix is defined by its dimensions, where $m$ represents the number of rows and $n$ represents the number of columns. The total number of elements $N$ in such a matrix is given by the product of the number of rows and the number of columns: $N = m \times n$ [By the Definition of Matrix Order, where $m, n \in \mathbb{N}$]. To find all possible orders for a matrix with $N$ elements, we must identify all possible ordered pairs $(m, n)$ such that their product equals $N$.

Step 2: Analysis for 18 Elements

Given $N = 18$, we seek all pairs of natural numbers $(m, n)$ such that $m \times n = 18$. We identify these by listing the factors of 18:

• $1 \times 18 = 18$
• $2 \times 9 = 18$
• $3 \times 6 = 18$
• $6 \times 3 = 18$
• $9 \times 2 = 18$
• $18 \times 1 = 18$

Step 3: Analysis for 5 Elements

Given $N = 5$, we identify the factors of 5. Since 5 is a prime number [By the Fundamental Theorem of Arithmetic], its only divisors are 1 and 5.

• $1 \times 5 = 5$
• $5 \times 1 = 5$

Step 4: Conclusion

The possible orders are determined by the set of all divisor pairs of the given integer $N$.

Final Solution: For 18 elements, the possible orders are $1 \times 18$, $2 \times 9$, $3 \times 6$, $6 \times 3$, $9 \times 2$, and $18 \times 1$. For 5 elements, the possible orders are $1 \times 5$ and $5 \times 1$.

Step 1: Establishing the System of Linear Equations

By the definition of equality of matrices, two matrices are equal if and only if their corresponding elements are identical [Equality of Matrices Definition]. Given the column matrices of order $3 \times 1$, we equate the individual elements to form a system of three linear equations:

• Eq (1): $x + y + z = 9$
• Eq (2): $x + z = 5$
• Eq (3): $y + z = 7$

Step 2: Solving for Variable $y$

Substitute Eq (2) into Eq (1) to isolate the variable $y$:

$(x + z) + y = 9$

$5 + y = 9$ [Substituting $x + z = 5$ from Eq (2)]

$y = 9 - 5$

$y = 4$

Step 3: Solving for Variable $x$

Substitute the value of $y = 4$ into Eq (3) to find $z$:

$4 + z = 7$ [Substituting $y = 4$ into Eq (3)]

$z = 7 - 4$

$z = 3$

Step 4: Solving for Variable $x$

Substitute the value of $z = 3$ into Eq (2) to find $x$:

$x + 3 = 5$ [Substituting $z = 3$ into Eq (2)]

$x = 5 - 3$

$x = 2$

Step 5: Verification of Results

We verify the values by substituting $x=2, y=4, z=3$ into Eq (1):

$2 + 4 + 3 = 9$

$9 = 9$ [LHS = RHS; Solution is consistent]

Visual Representation of Matrix Equality

Final Solution: $x = 2, y = 4, z = 3$

Step 1: Establishing Equality of Matrices

Two matrices are considered equal if and only if they have the same order and their corresponding elements are identical [Definition of Matrix Equality]. Given the equation:

$\begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$

By equating the corresponding elements, we derive the following system of non-linear equations:

• $x + y = 6$ (Eq. 1)
• $5 + z = 5$ (Eq. 2)
• $xy = 8$ (Eq. 3)

Step 2: Solving for $z$

From (Eq. 2), we isolate $z$:

$5 + z = 5$

$z = 5 - 5$

$z = 0$ [Additive Inverse Property]

Step 3: Solving the System for $x$ and $y$

We use (Eq. 1) to express $y$ in terms of $x$:

$y = 6 - x$ (Eq. 4)

Substitute (Eq. 4) into (Eq. 3):

$x(6 - x) = 8$

$6x - x^2 = 8$

$x^2 - 6x + 8 = 0$ [Rearranging into standard Quadratic Form $ax^2 + bx + c = 0$]

Applying the factorization method: find two numbers that multiply to $8$ and add to $-6$. These numbers are $-2$ and $-4$.

$(x - 2)(x - 4) = 0$

This yields two potential values for $x$:

$x = 2$ or $x = 4$

Step 4: Determining Corresponding $y$ Values

Using (Eq. 4) $y = 6 - x$:

Final Solution:

The values are $x = 2, y = 4, z = 0$ OR $x = 4, y = 2, z = 0$.

Step 1: Analyzing the Matrix Dimensions

Given the matrix $A$:

$A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$

To determine the number of elements, we first identify the dimensions (order) of the matrix. The order of a matrix is defined as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns [Definition of Matrix Order].

• Number of rows ($m$): There are 3 horizontal lines of elements. Thus, $m = 3$.
• Number of columns ($n$): There are 4 vertical lines of elements. Thus, $n = 4$.

Therefore, the order of matrix $A$ is $3 \times 4$.

Step 2: Calculating Total Elements

The total number of elements in a matrix $A$ of order $m \times n$ is given by the product of the number of rows and the number of columns [Fundamental Counting Principle].

$\text{Total elements} = m \times n$

Substituting the values derived in Step 1:

$\text{Total elements} = 3 \times 4 = 12$

Visual Representation of Matrix Grid

Final Solution: The number of elements in the given matrix is 12.

Step 1: Theoretical Definition of Matrix Elements

An element of a matrix is denoted by $a_{ij}$, where $i$ represents the row index and $j$ represents the column index. The index $i$ corresponds to the horizontal arrangement, and $j$ corresponds to the vertical arrangement of the element within the grid [Per the Definition of Matrix Notation].

Step 2: Analysis of the Given Matrix Structure

The matrix $A$ is provided as:

Step 3: Extraction of Requested Elements

By mapping the indices $(i, j)$ to the corresponding numerical values in the matrix, we extract the required components:

• Element $a_{13}$: Located in the 1st row and 3rd column. Value: $19$.
• Element $a_{21}$: Located in the 2nd row and 1st column. Value: $35$.
• Element $a_{33}$: Located in the 3rd row and 3rd column. Value: $-5$.
• Element $a_{24}$: Located in the 2nd row and 4th column. Value: $12$.
• Element $a_{23}$: Located in the 2nd row and 3rd column. Value: $\frac{5}{2}$.

Step 4: Data Tabulation

Final Solution: $a_{13} = 19, a_{21} = 35, a_{33} = -5, a_{24} = 12, a_{23} = \frac{5}{2}$

Step 1: Theoretical Foundation of Matrix Order

An $m \times n$ matrix is defined by $m$ rows and $n$ columns. The total number of elements in such a matrix is given by the product of the number of rows and the number of columns, denoted as $N = m \times n$ [Per the definition of the order of a matrix]. To find all possible orders for a matrix with $N$ elements, we must identify all unique factor pairs $(m, n)$ of the integer $N$, where $m, n \in \mathbb{N}$.

Step 2: Analysis for 24 Elements

To find the possible orders for $N = 24$, we determine all positive integer factor pairs of 24. We systematically test divisors of 24:

• $24 \div 1 = 24 \implies (1 \times 24)$ and $(24 \times 1)$
• $24 \div 2 = 12 \implies (2 \times 12)$ and $(12 \times 2)$
• $24 \div 3 = 8 \implies (3 \times 8)$ and $(8 \times 3)$
• $24 \div 4 = 6 \implies (4 \times 6)$ and $(6 \times 4)$

Step 3: Analysis for 13 Elements

Since 13 is a prime number, its only divisors are 1 and itself [Per the Fundamental Theorem of Arithmetic]. Therefore, the only possible factor pairs are:

• $13 \div 1 = 13 \implies (1 \times 13)$ and $(13 \times 1)$

Step 4: Geometric Representation of Matrix Structure

Final Solution: For 24 elements, the possible orders are: $1 \times 24, 24 \times 1, 2 \times 12, 12 \times 2, 3 \times 8, 8 \times 3, 4 \times 6, 6 \times 4$. For 13 elements, the possible orders are: $1 \times 13$ and $13 \times 1$.

Step 1: Establishing the Matrix Framework

For a $2 \times 2$ matrix $A$, the general representation is defined as:

Where $i$ represents the row index and $j$ represents the column index [Per the definition of Matrix Dimensions $m \times n$].

Step 2: Applying the Element Calculation Formula

Given the formula $a_{ij} = \frac{i}{j}$, we calculate each element systematically:

• For $i = 1, j = 1$: $a_{11} = \frac{1}{1} = 1$
• For $i = 1, j = 2$: $a_{12} = \frac{1}{2} = 0.5$
• For $i = 2, j = 1$: $a_{21} = \frac{2}{1} = 2$
• For $i = 2, j = 2$: $a_{22} = \frac{2}{2} = 1$

Step 3: Constructing the Final Matrix

Substituting the calculated values into the general $2 \times 2$ matrix structure:

Visual Representation of Matrix Elements

Final Solution: $A = \begin{bmatrix} 1 & \frac{1}{2} \\ 2 & 1 \end{bmatrix}$

Step 1: Definition of the $2 \times 2$ Matrix Structure

A matrix $A$ of order $2 \times 2$ is defined by its rows ($i$) and columns ($j$), where $i \in \{1, 2\}$ and $j \in \{1, 2\}$. The general form is represented as:

$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

Step 2: Element-wise Calculation using the Given Formula

The given formula for the elements is $a_{ij} = \frac{(i+2j)^2}{2}$. We calculate each element individually:

• For $a_{11}$ ($i=1, j=1$):
  $a_{11} = \frac{(1 + 2(1))^2}{2} = \frac{(1+2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5$ [Per the definition of the arithmetic operator hierarchy]
• For $a_{12}$ ($i=1, j=2$):
  $a_{12} = \frac{(1 + 2(2))^2}{2} = \frac{(1+4)^2}{2} = \frac{5^2}{2} = \frac{25}{2} = 12.5$
• For $a_{21}$ ($i=2, j=1$):
  $a_{21} = \frac{(2 + 2(1))^2}{2} = \frac{(2+2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8$
• For $a_{22}$ ($i=2, j=2$):
  $a_{22} = \frac{(2 + 2(2))^2}{2} = \frac{(2+4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18$

Step 3: Constructing the Matrix

Substituting the calculated values into the general $2 \times 2$ matrix structure:

$A = \begin{bmatrix} 4.5 & 12.5 \\ 8 & 18 \end{bmatrix}$

Visual Representation of Matrix Indices

Final Solution: $A = \begin{bmatrix} 4.5 & 12.5 \\ 8 & 18 \end{bmatrix}$

Step 1: Definition of a Matrix Order

In matrix algebra, the dimension or order of a matrix is denoted by $m \times n$, where:

• $m$ represents the number of horizontal rows.
• $n$ represents the number of vertical columns.

For any arbitrary matrix $A = [a_{ij}]_{m \times n}$, the element $a_{ij}$ occupies the position at the intersection of the $i$-th row and $j$-th column [Per the formal definition of matrix notation].

Step 2: Defining a Square Matrix

A matrix is classified as a square matrix if and only if the number of rows is exactly equal to the number of columns. This condition ensures that the matrix has an equal span in both dimensions, forming a geometric square in terms of its element distribution.

Step 3: Analytical Derivation

Given the general order $m \times n$:

• Let the condition for a square matrix be $m = n$.
• If $m < n$, the matrix is a horizontal rectangular matrix.
• If $m > n$, the matrix is a vertical rectangular matrix.
• If $m = n$, the matrix is a square matrix of order $m$ (or $n$).

Step 4: Visual Representation

Step 5: Theoretical Conclusion

By the definition of square matrices in linear algebra, the condition $m = n$ is the necessary and sufficient requirement for a matrix to be considered square. This implies that the matrix is an $n \times n$ matrix.

Final Solution: $m = n$

Step 1: Theoretical Foundation of the Matrix

A $3 \times 4$ matrix $A$ consists of $3$ rows and $4$ columns. The general representation of this matrix is:

$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{bmatrix}$

Each element $a_{ij}$ is determined by the mapping function $a_{ij} = \frac{1}{2}|-3i + j|$, where $i \in \{1, 2, 3\}$ and $j \in \{1, 2, 3, 4\}$.

Step 2: Systematic Calculation of Elements

We evaluate the elements row by row using the defined modulus function [Definition of Absolute Value: $|x| = x$ if $x \ge 0$, and $-x$ if $x < 0$].

Step 3: Matrix Assembly

Substituting the calculated values into the matrix structure defined in Step 1:

$A = \begin{bmatrix} 1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2} \end{bmatrix}$

Step 4: Geometric Representation of Matrix Indices

Final Solution: $A = \begin{bmatrix} 1 & 0.5 & 0 & 0.5 \\ 2.5 & 2 & 1.5 & 1 \\ 4 & 3.5 & 3 & 2.5 \end{bmatrix}$

Step 1: Establishing the System of Linear Equations

By the definition of equality of matrices, two matrices are equal if and only if their corresponding elements are identical [Per Definition of Matrix Equality]. Given the equation:

$\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$

We equate the corresponding entries to form a system of four linear equations:

• Eq (i): $a - b = -1$
• Eq (ii): $2a + c = 5$
• Eq (iii): $2a - b = 0$
• Eq (iv): $3c + d = 13$

Step 2: Solving for Variables $a$ and $b$

Utilizing Eq (i) and Eq (iii), we perform elimination [By Cramer’s Rule/Elimination Method]:

From Eq (iii), we isolate $b$: $b = 2a$.

Substitute $b = 2a$ into Eq (i):

$a - (2a) = -1$

$-a = -1 \implies a = 1$

Now, substitute $a = 1$ back into the expression for $b$:

$b = 2(1) = 2$

Step 3: Solving for Variables $c$ and $d$

Substitute $a = 1$ into Eq (ii) to determine $c$:

$2(1) + c = 5$

$2 + c = 5 \implies c = 3$

Now, substitute $c = 3$ into Eq (iv) to determine $d$:

$3(3) + d = 13$

$9 + d = 13$

$d = 13 - 9 = 4$

Visual Representation of Matrix Mapping

Step 4: Summary Table of Results

Final Solution: The values are $a=1, b=2, c=3, d=4$.