Comprehensive guide to matrix operations and properties in linear algebra.

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- Transpose of a matrix
- Symmetric and Skew-symmetric matrices
- Determinant of a matrix
- Area of a triangle using determinants
- Adjoint of a square matrix

However, there is no specific problem or question stated in the image. You mentioned "I uploaded an image with a task," but the content appears to be reference material, not a problem to solve.

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✔ What I Can Do:

If your goal was to understand these concepts, here’s a brief explanation of each section for clarity:

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1. Transpose of a Matrix

Given a matrix $ A = [a_{ij}]_{m \times n} $, its transpose $ A^T $ is obtained by swapping rows and columns.

> Example:
> If
> $$
> A = \begin{bmatrix}
> 1 & 2 \\
> 3 & 4
> \end{bmatrix}, \quad
> A^T = \begin{bmatrix}
> 1 & 3 \\
> 2 & 4
> \end{bmatrix}
> $$

#### Properties:
- $ (A^T)^T = A $
- $ (A + B)^T = A^T + B^T $
- $ (kA)^T = kA^T $
- $ (AB)^T = B^T A^T $

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2. Symmetric Matrix

A square matrix $ A $ is symmetric if $ a_{ij} = a_{ji} $ for all $ i, j $, or equivalently $ A^T = A $.

> Example:
> $$
> A = \begin{bmatrix}
> 1 & 2 \\
> 2 & 3
> \end{bmatrix} \quad \text{is symmetric}
> $$

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3. Skew-Symmetric Matrix

A square matrix $ A $ is skew-symmetric if $ a_{ij} = -a_{ji} $ for all $ i, j $, or $ A^T = -A $. Diagonal elements must be zero.

> Example:
> $$
> A = \begin{bmatrix}
> 0 & -2 \\
> 2 & 0
> \end{bmatrix} \quad \text{is skew-symmetric}
> $$

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4. Determinant

For a square matrix $ A $ of order $ n \times n $, the determinant is denoted $ |A| $ or $ \det(A) $.

#### Key Properties:
- $ |AB| = |A||B| $
- $ |kA| = k^n |A| $ for scalar $ k $

#### Area of Triangle:
Given three points $ (x_1, y_1), (x_2, y_2), (x_3, y_3) $, the area is:
$$
\Delta = \frac{1}{2} \left| \begin{array}{ccc}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array} \right|
$$

The points are collinear iff this determinant is zero.

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5. Adjoint of a Matrix

For a square matrix $ A $, the adjoint $ \text{adj}(A) $ is the transpose of the cofactor matrix.

Let $ C_{ij} $ be the cofactor of element $ a_{ij} $. Then:
$$
\text{adj}(A) = [C_{ji}] \quad \text{(transpose of cofactor matrix)}
$$

> Note: $ A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A) $, provided $ |A| \ne 0 $

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