Unit 1: Matrices | ફન લર્નિંગ શ્રેણિકો

1.1 Concept of Matrix (શ્રેણિકનો ખ્યાલ)

સંખ્યાઓની લંબચોરસ ગોઠવણી કે જેને [ ] અથવા ( ) કૌંસમાં દર્શાવવામાં આવે છે તેને શ્રેણિક કહે છે. ઘટકોની આડી રેખાઓને હાર (Rows) અને ઊભી રેખાઓને સ્તંભ (Columns) કહે છે.

Order of Matrix (શ્રેણિકની કક્ષા): If a matrix has $m$ rows and $n$ columns, its order is written as $m \times n$.

Equality of Matrices (શ્રેણિકોની સમાનતા): Two matrices are equal if their orders are the same and their corresponding elements are exactly equal.

✍️ Example 1: Order of Matrix (Click to view)

Question: What is the order of $A = \begin{bmatrix} 1 & 0 & 3 \\ -2 & 4 & 0 \end{bmatrix}$?

Rows (હાર) = 2, Columns (સ્તંભ) = 3.
Answer: The order is $2 \times 3$.

✍️ Example 2: Order of Matrix (Click to view)

Question: What is the order of $A = \begin{bmatrix} 2 & 5 \\ 7 & 8 \end{bmatrix}$?

Rows (હાર) = 2, Columns (સ્તંભ) = 2.
Answer: The order is $2 \times 2$.

✍️ Example 3: Order of Matrix (Click to view)

Question: What is the order of $A = \begin{bmatrix} 1 & 2 \\ 0 & -1 \\ 3 & 4 \end{bmatrix}$?

Rows (હાર) = 3, Columns (સ્તંભ) = 2.
Answer: The order is $3 \times 2$.

✍️ Example 4: Equality of Matrices (Click to view)

Question: If $\begin{bmatrix} x+y & 3 \\ -7 & x-y \end{bmatrix} = \begin{bmatrix} 8 & 3 \\ -7 & 2 \end{bmatrix}$, find $x$ and $y$.

Compare elements: $x + y = 8$ and $x - y = 2$.
Add them: $2x = 10 \Rightarrow x = 5$.
Substitute: $5 + y = 8 \Rightarrow y = 3$.
Answer: $x = 5, y = 3$.

1.2 Types of Matrices (શ્રેણિકોના પ્રકારો)

શ્રેણિકોના વિવિધ પ્રકારો (Different types of matrices).

Square Matrix (ચોરસ શ્રેણિક): Number of rows = Number of columns.
Identity Matrix (એકમ શ્રેણિક): Principal diagonal elements are 1, others are 0. Denoted by $I$.
Transpose (પરિવર્ત શ્રેણિક $A^T$): Swap all rows into corresponding columns (હારને સ્તંભમાં ફેરવતા).
Singular Matrix (અસામાન્ય શ્રેણિક): Determinant is zero ($|A| = 0$).

✍️ Example 1: Square Matrix (Click to view)

Question: Which of the following is a square matrix? a) $[1, 3]$ b) $\begin{bmatrix} 1 \\ 5 \end{bmatrix}$ c) $\begin{bmatrix} 1 & 3 \\ 5 & 4 \end{bmatrix}$ d) $\begin{bmatrix} 1 & 3 & 2 \\ 5 & 4 & 5 \end{bmatrix}$

A square matrix has equal rows and columns.
Answer: Option (c) because it is a $2 \times 2$ matrix.

✍️ Example 2: Transpose (Click to view)

Question: If $A = \begin{bmatrix} 1 & -1 \end{bmatrix}$, then $A^T = ?$

Change the row into a column.
Answer: $A^T = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$.

✍️ Example 3: Transpose variables (Click to view)

Question: If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $A^T = ?$

Swap rows to columns. Row 1 $[a, b]$ becomes Column 1.
Answer: $A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$.

✍️ Example 4: Singular Matrix (Click to view)

Question: If $A$ is a non-singular matrix, then... ?

For a matrix to be non-singular (સામાન્ય શ્રેણિક), its determinant must not be zero.
Answer: $|A| \neq 0$.

1.3 Addition, Subtraction & Scalar Multiplication (સરવાળા, બાદબાકી અને અદિશ ગુણાકાર)

શ્રેણિકોના સરવાળા અને બાદબાકી ત્યારે જ શક્ય બને છે જો બંને શ્રેણિકોની કક્ષા સમાન હોય.

Add/subtract corresponding elements. For scalar multiplication, multiply every single element inside the matrix by the given number.

✍️ Example 1: Scalar Multiplication (Click to view)

Question: If $A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}$, find $4A$.

Multiply every element by 4.
Answer: $4A = \begin{bmatrix} 4 & 8 \\ 12 & -4 \end{bmatrix}$.

✍️ Example 2: Scalar Math (Click to view)

Question: If $A = \begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix}$, find $2A - 3I$.

$2A = \begin{bmatrix} 2 & 4 \\ -2 & 2 \end{bmatrix}$. $3I = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$.
Subtract: $\begin{bmatrix} 2-3 & 4-0 \\ -2-0 & 2-3 \end{bmatrix}$.
Answer: $\begin{bmatrix} -1 & 4 \\ -2 & -1 \end{bmatrix}$.

✍️ Example 3: Addition (Click to view)

Question: Find $\begin{bmatrix} 4 & 3 \\ 6 & 2 \end{bmatrix} + \begin{bmatrix} 1 & 5 \\ 5 & 8 \end{bmatrix}$.

Add matching positions.
Answer: $\begin{bmatrix} 4+1 & 3+5 \\ 6+5 & 2+8 \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 11 & 10 \end{bmatrix}$.

✍️ Example 4: Algebra (Click to view)

Question: If $A = \begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix}$, find $2A - 2I$.

$2A = \begin{bmatrix} 2 & 6 \\ 8 & -4 \end{bmatrix}$. $2I = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$.
Answer: $\begin{bmatrix} 0 & 6 \\ 8 & -6 \end{bmatrix}$.

1.4 Product of Two Matrices (બે શ્રેણિકોનો ગુણાકાર)

ગુણાકાર AB ત્યારે જ શક્ય છે જો A ના સ્તંભની સંખ્યા = B ની હારની સંખ્યા હોય.

Multiply the rows of the first matrix by the corresponding columns of the second matrix.

✍️ Example 1: Basic Product (Click to view)

Question: If $A = \begin{bmatrix} 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \end{bmatrix}$, then $A \cdot B = ?$

Order of A is $1 \times 1$, Order of B is $1 \times 1$.
Answer: $\begin{bmatrix} 3 \times 4 \end{bmatrix} = \begin{bmatrix} 12 \end{bmatrix}$.

✍️ Example 2: 2x2 Multiplication (Click to view)

Question: Find $\begin{bmatrix} 1 & 2 \\ 5 & 0 \end{bmatrix} \times \begin{bmatrix} -1 & 6 \\ 2 & 1 \end{bmatrix}$.

$R_1 C_1: (1)(-1) + (2)(2) = 3$
$R_1 C_2: (1)(6) + (2)(1) = 8$
$R_2 C_1: (5)(-1) + (0)(2) = -5$
$R_2 C_2: (5)(6) + (0)(1) = 30$
Answer: $\begin{bmatrix} 3 & 8 \\ -5 & 30 \end{bmatrix}$.

✍️ Example 3: Finding Unknowns in Product (Click to view)

Question: If $\begin{bmatrix} 0 & x & -2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \end{bmatrix}$, find $x$.

Multiply: $(0)(1) + (x)(2) + (-2)(3) = 4$
$0 + 2x - 6 = 4 \Rightarrow 2x = 10$.
Answer: $x = 5$.

✍️ Example 4: Order of Product (Click to view)

Question: If A is a $2 \times 3$ matrix and B is a $3 \times 4$ matrix, what is the order of AB?

The inner dimensions (3 and 3) match. The result takes the outer dimensions.
Answer: The order of AB is $2 \times 4$.

1.4a Mixed Examples (મિશ્ર ઉદાહરણો 1.1 થી 1.4)

સરવાળા, બાદબાકી, અદિશ ગુણાકાર, ગુણાકાર અને પરિવર્ત શ્રેણિક ના મિશ્ર દાખલા.

✍️ Example 1: Transpose Rule (Click to view)

Question: If $A = \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -1 \\ 2 & 3 \end{bmatrix}$, prove $(A+B)^T = A^T + B^T$.

L.H.S: $A+B = \begin{bmatrix} 6 & 0 \\ 5 & 3 \end{bmatrix} \Rightarrow (A+B)^T = \begin{bmatrix} 6 & 5 \\ 0 & 3 \end{bmatrix}$.

R.H.S: $A^T = \begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix}$, $B^T = \begin{bmatrix} 4 & 2 \\ -1 & 3 \end{bmatrix}$. Add them: $A^T+B^T = \begin{bmatrix} 6 & 5 \\ 0 & 3 \end{bmatrix}$.
Answer: L.H.S = R.H.S. Proved.

✍️ Example 2: Combined Operations (Click to view)

Question: If $A = \begin{bmatrix} 2 & 5 \\ -1 & 3 \end{bmatrix}, B = \begin{bmatrix} 5 & 8 \\ 4 & 6 \end{bmatrix}, C = \begin{bmatrix} 4 & 2 \\ 1 & 5 \end{bmatrix}$, find $2A + 3B - C$.

$2A = \begin{bmatrix} 4 & 10 \\ -2 & 6 \end{bmatrix}, 3B = \begin{bmatrix} 15 & 24 \\ 12 & 18 \end{bmatrix}$.

Add $2A + 3B = \begin{bmatrix} 19 & 34 \\ 10 & 24 \end{bmatrix}$.
Subtract C: $\begin{bmatrix} 19-4 & 34-2 \\ 10-1 & 24-5 \end{bmatrix}$.
Answer: $\begin{bmatrix} 15 & 32 \\ 9 & 19 \end{bmatrix}$.

✍️ Example 3: Finding Unknown Matrix (Click to view)

Question: If $A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$, find matrix $X$ such that $2A + X = 3B$.

Rearrange formula: $X = 3B - 2A$.
$3B = \begin{bmatrix} 9 & 6 \\ 3 & 12 \end{bmatrix}$, $2A = \begin{bmatrix} 4 & -2 \\ 8 & 6 \end{bmatrix}$.

$X = \begin{bmatrix} 9-4 & 6-(-2) \\ 3-8 & 12-6 \end{bmatrix}$.
Answer: $X = \begin{bmatrix} 5 & 8 \\ -5 & 6 \end{bmatrix}$.

✍️ Example 4: Proving Matrix Equation (Click to view)

Question: If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = 0$.

$A^2 = A \times A = \begin{bmatrix} 9-1 & 3+2 \\ -3-2 & -1+4 \end{bmatrix} = \begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix}$.

$-5A = \begin{bmatrix} -15 & -5 \\ 5 & -10 \end{bmatrix}$ and $7I = \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$.

Add them: $\begin{bmatrix} 8-15+7 & 5-5+0 \\ -5+5+0 & 3-10+7 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Proved.

1.5 Adjoint & Inverse of 2x2 and 3x3 Matrices

ચોરસ શ્રેણિક A ના વ્યસ્ત શ્રેણિકને $A^{-1}$ વડે દર્શાવવામાં આવે છે. સૂત્ર: $A^{-1} = \frac{1}{|A|} adj A$

✍️ Example 1: Adjoint of 2x2 (Click to view)

Question: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $adj A$.

Shortcut: Swap the principal diagonal (1 & 4), change signs of the other diagonal (2 & 3).
Answer: $adj A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$.

✍️ Example 2: Inverse of 2x2 (Click to view)

Question: Find the inverse of $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$.

Determinant $|A| = (3 \times 2) - (1 \times -1) = 6 + 1 = 7$.

Adjoint $adj A = \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}$.
Answer: $A^{-1} = \frac{1}{7} \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}$.

✍️ Example 3: Adjoint Trick (Click to view)

Question: If $A = \begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}$, then $adj A = ?$

Swap diagonal elements (1 and -1), flip signs of (-2 and 2).
Answer: $adj A = \begin{bmatrix} -1 & 2 \\ -2 & 1 \end{bmatrix}$.

✍️ Example 4: Inverse of 3x3 (Click to view)

Question: Find $A^{-1}$ for $A = \begin{bmatrix} 3 & -1 & 2 \\ 4 & 1 & -1 \\ 5 & 0 & 1 \end{bmatrix}$.

Step 1: Determinant $|A| = 3(1-0) - (-1)(4-(-5)) + 2(0-5) = 3 + 9 - 10 = 2$.

Step 2: Cofactor Matrix = $\begin{bmatrix} 1 & -9 & -5 \\ 1 & -7 & -5 \\ -1 & 11 & 7 \end{bmatrix}$.

Step 3: Transpose to get $adj A = \begin{bmatrix} 1 & 1 & -1 \\ -9 & -7 & 11 \\ -5 & -5 & 7 \end{bmatrix}$.
Answer: $A^{-1} = \frac{1}{2} \begin{bmatrix} 1 & 1 & -1 \\ -9 & -7 & 11 \\ -5 & -5 & 7 \end{bmatrix}$.

1.6 Solution of Simultaneous Linear Equations

શ્રેણિકની મદદથી સમીકરણો ઉકેલવા: $AX = K \Rightarrow X = A^{-1}K$

✍️ Example 1: Matrix Method (Click to view)

Question: Solve using matrices: $3x - 2y = 8$ and $5x + 4y = 6$.

Form $AX=K \Rightarrow A = \begin{bmatrix} 3 & -2 \\ 5 & 4 \end{bmatrix}, K = \begin{bmatrix} 8 \\ 6 \end{bmatrix}$.

$|A| = 22$. $adj A = \begin{bmatrix} 4 & 2 \\ -5 & 3 \end{bmatrix}$. $A^{-1} = \frac{1}{22} \begin{bmatrix} 4 & 2 \\ -5 & 3 \end{bmatrix}$.

$X = A^{-1}K = \frac{1}{22} \begin{bmatrix} 32+12 \\ -40+18 \end{bmatrix} = \frac{1}{22} \begin{bmatrix} 44 \\ -22 \end{bmatrix}$.
Answer: $x=2, y=-1$.

✍️ Example 2: Matrix Method (Click to view)

Question: Solve $2x + 3y = 8$ and $3x + 4y = 11$.

Form $A = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}, K = \begin{bmatrix} 8 \\ 11 \end{bmatrix}$.

$|A| = (8 - 9) = -1$. $A^{-1} = \frac{1}{-1} \begin{bmatrix} 4 & -3 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} -4 & 3 \\ 3 & -2 \end{bmatrix}$.

$X = \begin{bmatrix} -4 & 3 \\ 3 & -2 \end{bmatrix} \begin{bmatrix} 8 \\ 11 \end{bmatrix} = \begin{bmatrix} -32+33 \\ 24-22 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$.
Answer: $x=1, y=2$.

✍️ Example 3: Matrix Method (Click to view)

Question: Solve $3x + 2y = 5$ and $2x - y = 1$.

Form $A = \begin{bmatrix} 3 & 2 \\ 2 & -1 \end{bmatrix}, K = \begin{bmatrix} 5 \\ 1 \end{bmatrix}$.

$|A| = (-3 - 4) = -7$. $A^{-1} = \frac{1}{-7} \begin{bmatrix} -1 & -2 \\ -2 & 3 \end{bmatrix}$.

$X = \frac{1}{-7} \begin{bmatrix} -1 & -2 \\ -2 & 3 \end{bmatrix} \begin{bmatrix} 5 \\ 1 \end{bmatrix} = \frac{1}{-7} \begin{bmatrix} -7 \\ -7 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
Answer: $x=1, y=1$.

✍️ Example 4: Arranging Equations (Click to view)

Question: Solve $2y + 5x = 4$ and $7x + 3y = 5$.

Rewrite properly: $5x + 2y = 4$ and $7x + 3y = 5$.
$A = \begin{bmatrix} 5 & 2 \\ 7 & 3 \end{bmatrix}, K = \begin{bmatrix} 4 \\ 5 \end{bmatrix}$.

$|A| = 15 - 14 = 1$. $A^{-1} = \begin{bmatrix} 3 & -2 \\ -7 & 5 \end{bmatrix}$.

$X = \begin{bmatrix} 3 & -2 \\ -7 & 5 \end{bmatrix} \begin{bmatrix} 4 \\ 5 \end{bmatrix} = \begin{bmatrix} 12-10 \\ -28+25 \end{bmatrix} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}$.
Answer: $x=2, y=-3$.

🚀 The Matrix Master Lab (સ્ટેપ-બાય-સ્ટેપ કેલ્ક્યુલેટર)

Select dimensions, enter numbers, and get detailed step-by-step solutions! / કક્ષા પસંદ કરો, સંખ્યા દાખલ કરો અને સ્ટેપ-બાય-સ્ટેપ ઉકેલ મેળવો!

Matrix A: ×

Matrix B: ×

Matrix A

Matrix B

Select an operation above to see the step-by-step solution here! / ઉકેલ જોવા માટે ઉપરનું કોઈપણ બટન દબાવો.

📝 Linear Equation Solver (સુરેખ સમીકરણ ઉકેલ)

Enter your pair of linear equations / સમીકરણો દાખલ કરો:

Equation 1: $x +$ $y =$

Equation 2: $x +$ $y =$

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