Matrices

Class 12 · Mathematics

Matrices

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Matrices

Ek matrix numbers ki rectangular table hai (rows aur columns me). Ye bahut saari information ko compact tareeke se rakhne aur linear equations ko handle karne ka powerful tool hai.

1. Matrix aur uska Order

Numbers ko rows aur columns me arrange karna = matrix. \(m\) rows aur \(n\) columns wali matrix ka order \(m \times n\) hota hai.

Example (order \(2 \times 3\)):

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]

Element \(a_{ij}\) = \(i\)-th row, \(j\)-th column ka number. Yahan \(a_{23} = 6\).

2. Types of Matrices

  • Row matrix: sirf ek row.
  • Column matrix: sirf ek column.
  • Square matrix: rows \(=\) columns.
  • Diagonal matrix: diagonal ke alawa sab \(0\).
  • Identity matrix \(I\): diagonal par \(1\), baaki \(0\).
  • Zero matrix \(O\): saare elements \(0\).

3. Operations on Matrices

Equality: do matrices equal tabhi jab same order aur har corresponding element same ho.

Addition / Subtraction: same order ki matrices me corresponding elements jodo/ghatao.

Scalar multiplication: har element ko scalar \(k\) se multiply karo.

Matrix Multiplication

\(A\) (order \(m \times n\)) aur \(B\) (order \(n \times p\)) ka product \(AB\) order \(m \times p\) ka hota hai. Multiply tabhi possible jab \(A\) ke columns \(=\) \(B\) ke rows.

\((AB)_{ij}\) = \(A\) ki \(i\)-th row aur \(B\) ke \(j\)-th column ka "dot product".

Important: matrix multiplication commutative nahi hota — aam taur par \(AB \neq BA\).

Example

\[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1\cdot5 + 2\cdot7 & 1\cdot6 + 2\cdot8 \\ 3\cdot5 + 4\cdot7 & 3\cdot6 + 4\cdot8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}. \]

4. Transpose

Matrix \(A\) ka transpose \(A^{T}\) — rows aur columns ko aapas me badal do.

Properties: \((A^{T})^{T} = A\), \((A + B)^{T} = A^{T} + B^{T}\), \((AB)^{T} = B^{T}A^{T}\) (order ulta!).

5. Symmetric aur Skew-Symmetric

  • Symmetric: \(A^{T} = A\).
  • Skew-symmetric: \(A^{T} = -A\) (iske diagonal elements hamesha \(0\) hote hain).

Key Takeaways

  • Matrix ka order \(m \times n\); element \(a_{ij}\).
  • Add/subtract ke liye same order chahiye; scalar har element par lagta hai.
  • Multiply karne ke liye \(A\) ke columns \(=\) \(B\) ke rows; \(AB \neq BA\) (commutative nahi).
  • Transpose: rows↔columns; \((AB)^{T} = B^{T}A^{T}\).
  • Symmetric: \(A^{T} = A\); skew-symmetric: \(A^{T} = -A\).
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