| --- |
| source: ncert |
| topic: algebra |
| class_level: class_12 |
| chapter: ch3 |
| difficulty: advanced |
| --- |
| |
| Class 12 | Ch3: Matrices |
| MATRIX: rectangular array of numbers in rows and columns. Order: m×n (m rows, n cols). |
| TYPES: Row(1×n), Column(m×1), Square(n×n), Zero/Null, Identity(diagonal=1, rest=0), Diagonal, Scalar. |
| OPERATIONS: |
| Addition: (A+B)_ij=a_ij+b_ij. Same order required. |
| Scalar multiplication: (kA)_ij=k×a_ij. |
| Multiplication: (AB)_ij=sum over k of a_ik×b_kj. Order: (m×n)(n×p)=(m×p). |
| AB is defined iff columns of A=rows of B. AB≠BA generally. |
| TRANSPOSE: A^T: rows become columns. (A^T)_ij=A_ji. |
| (A+B)^T=A^T+B^T. (AB)^T=B^T A^T. (A^T)^T=A. |
| SYMMETRIC: A^T=A (a_ij=a_ji). SKEW-SYMMETRIC: A^T=-A (a_ij=-a_ji, diagonal=0). |
| ANY SQUARE MATRIX = Symmetric part + Skew-symmetric part. |
| A=(A+A^T)/2 + (A-A^T)/2. First part symmetric, second skew-symmetric. |
| ELEMENTARY OPERATIONS (Row/Column): |
| Ri↔Rj (interchange). Ri→kRi (multiply by scalar). Ri→Ri+kRj (add multiple of another row). |
| Used in finding inverse by row reduction. |
| SOLVED EXAMPLES: |
| Example 1: A=[[1,2],[3,4]]. Find A+A^T and show symmetric. |
| A^T=[[1,3],[2,4]]. A+A^T=[[2,5],[5,8]]. (A+A^T)^T=[[2,5],[5,8]]=A+A^T. Symmetric. |
| Example 2: Express A=[[1,2],[3,4]] as sum of symmetric and skew-symmetric. |
| Symmetric=(A+A^T)/2=[[1,2.5],[2.5,4]]. |
| Skew-symmetric=(A-A^T)/2=[[0,-0.5],[0.5,0]]. |
| Example 3: If A=[[1,2],[3,4]] and B=[[5,6],[7,8]], find AB. |
| AB=[[1×5+2×7, 1×6+2×8],[3×5+4×7, 3×6+4×8]]=[[19,22],[43,50]]. |
| COMMON MISTAKES: |
| AB≠BA in general (matrix multiplication is NOT commutative). |
| (AB)^T=B^T A^T (ORDER REVERSES, not A^T B^T). |
| Dimensions: (m×n)(n×p)→(m×p). Inner dimensions must match. |
