Definitions [10]
To find the determinant, multiply each element of your chosen row (or column) by its corresponding sign multiplier \[(-1)^{i+j}\] and the \[2 \times 2\] determinant that remains after deleting that element's row and column.
Let A = [aij] be a square matrix of order n. Then, the minor Mij of aij in A is the determinant obtained by deleting the ith row and the jth column in which element aij lies. It is denoted by Mij of A.
Let A = [aij] be a square matrix of order n. Then, the cofactor Cij (or Aij) of aij in A is (−1)i+j times Mij, where Mij is the minor of aij in A.
∴ Cij = (−1)i+j Mij
Consistent Solution: A system is consistent if it has at least one solution.
Inconsistent Solution: A system is inconsistent if it has no solution.
A determinant is a scalar value associated with a square matrix.
For a \[ 2 \times 2 \] matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
its determinant is written as: \[ |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]
and calculated as: \[ |A| = ad - bc \]
Remember: A determinant is a number, while a matrix is an arrangement of numbers.
A matrix is a rectangular arrangement of numbers arranged in rows and columns, enclosed in brackets [ ] or parentheses ( ).
Elements (Entries) of a Matrix
- Each number in a matrix is called an element (or entry).
Rows and Columns
- Horizontal lines → rows
- Vertical lines → columns
Order of a Matrix
- Order = number of rows × number of columns
- Written as m × n and read as “m by n”
Let \[A = [a_{ij}]_{m \times n}\] be a matrix and k be a real number (scalar).
Then the scalar multiple of A by k is the matrix kA defined as:
That is, each entry of A is multiplied by the scalar k.
The negative of a matrix A, denoted by -A, is defined as the scalar multiple \[-1 \cdot A\].
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So, if \[A = [a_{ij}]\], then \[-A = [-a_{ij}]\]
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Adding a matrix to its negative gives the zero matrix: A + (-A) = O
where O is the zero matrix of the same order as A.
Let \[A = [a_{ij}]\] be an \[m \times n\] matrix and \[B = [b_{jk}]\] be an \[n \times p\] matrix.
Then the product C = AB is an \[m \times p\] matrix \[C = [c_{ik}]\], where each entry \[c_{ik}\] is given by:
The transpose of a matrix is obtained by interchanging its rows and columns.
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If a matrix is A, its transpose is denoted by AT
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If A is of order m × n, then
AT is of order n × m - First row of A becomes first column of AT, and so on.
Formulae [2]
To expand along the first row, multiply each element of the first row by: \[ (-1)^{i+j} \] and by the second-order determinant obtained after deleting the row and column containing that element.
Thus,
\[ |A| = (-1)^{1+1}a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} + (-1)^{1+2}a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + (-1)^{1+3}a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}. \]
Since the signs are \[+, -, +\],
\[ |A| = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix} \]
or,
\[ |A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]
Expansion Along the Second Row \[(R_2)\]
The sign pattern along the second row is \[ -, +, -. \]
Therefore,
\[ |A| = -a_{21}\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} + a_{22}\begin{vmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{vmatrix} - a_{23}\begin{vmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{vmatrix}. \]
Expansion along \[R_2\] gives the same value as expansion along \[R_1\].
Expansion Along the First Column \[(C_1)\]
The signs down the first column are \[ +, -, +. \]
Thus,
\[ |A| = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{21}\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} + a_{31}\begin{vmatrix} a_{12} & a_{13} \\ a_{22} & a_{23} \end{vmatrix}. \]
Again, the final value is the same.
Cramer’s Rule
For the system: \[ a_1x + b_1y = c_1 \], \[ a_2x + b_2y = c_2 \]
\[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \]
\[ D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} \]
\[ D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \]
If \[ D \neq 0, \] then the system has a unique solution:
\[ x = \frac{D_x}{D} \]
\[ y = \frac{D_y}{D} \]
Key Points
| Concept | Formula/Rule |
|---|---|
| Expansion along R₁ | a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ |
| Expansion along C₁ | a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁ |
| Cofactor Sign | (-1)(i+j) → checkerboard: + - + / - + - / + - + |
| Zero Strategy | Expand along row/column with most zeros |
| Result Independence | Any row/column expansion gives same |
| Important Result | Order 3 determinant: 6 expansions (R1,R2,R3,C1,C2,C3),all give the same value |
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Minor \[M_{ij}\]: determinant of the matrix obtained by deleting row i and column j.
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Cofactor \[C_{ij}\]: \[C_{ij} = (-1)^{i+j}M_{ij}\].
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Determinant expansion along row i: \[|A| = \sum_{j=1}^{n} a_{ij}C_{ij}\].
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Determinant expansion along column j: \[|A| = \sum_{i=1}^{n} a_{ij}C_{ij}\].
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Determinant value is the same for any choice of row or column for expansion.
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Mixed row/column property: \[\sum_{j=1}^{n} a_{ij}C_{kj} = 0\] for \[i \neq k\].
| Concept | Key Point / Formula |
|---|---|
| Area of Triangle | \[ \boxed{\dfrac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}} \] |
| Collinearity | Three points are collinear if determinant =0=0 |
| Equation of Line | Line through two points can be written using a 3 × 3 determinant |
| Consistent System | Has at least one solution |
| Inconsistent System | Has no solution |
| Non-Singular Case | \[ |A| \neq 0 \] |
| Singular Case | ∣A∣=0 |
- A determinant is a number associated with a square matrix.
- For \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix}, \] the value is \[ ad - bc. \]
- Cramer’s Rule solves a pair of linear equations using determinants.
- \[ D \] is formed from the coefficients of \[ x \] and \[ y. \]
- \[ D_x \] replaces the \[ x\text{-coefficient} \] column with constants.
- \[ D_y \] replaces the \[ y\text{-coefficient} \] column with constants.
- If \[ D \neq 0, \] the solution is \[ x = \frac{D_x}{D}, y = \frac{D_y}{D}. \]
- If \[ D = 0, \] Cramer’s Rule does not give a unique solution.
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Matrix: A rectangular array of elements.
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Element: An entry inside a matrix.
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Order: Size of a matrix written as rows × columns.
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Row: Horizontal set of elements.
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Column: Vertical set of elements.
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aij: Element in the i-th row and j-th column.
| Matrix Type | Order | Key Property |
|---|---|---|
| Row Matrix | 1 × n | Only one row |
| Column Matrix | m × 1 | Only one column |
| Square Matrix | n × n | Rows = Column |
| Diagonal Matrix | n × n | Square; non-diagonal elements = 0 |
| Scalar Matrix | n × n | Diagonal; all diagonal elements equal |
| Identity Matrix | n × n | Scalar matrix with diagonal = 1 |
| Zero Matrix | Any order | All elements = 0 |
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Scalar multiplication: \[kA = [ka_{ij}]\].
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Negative of a matrix: -A = (-1)A.
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Order of matrix does not change after scalar multiplication.
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k(A + B) = kA + kB.
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(k + l)A = kA + lA.
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k(lA) = (kl)A.
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\[0 \cdot A = O\], \[1 \cdot A = A\].
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Matrix multiplication is row-by-column, not term-wise.
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Product AB exists only if columns of A = rows of B.
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If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].
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In general, \[AB \neq BA\], and sometimes one product may not even be defined.
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Matrix multiplication is associative and distributive over addition.
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Identity matrix acts as a multiplicative identity: AI = IA = A.
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Zero matrix absorbs multiplication: AO = OA = O.
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Transpose = interchange rows and columns.
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If A is \[m \times n\], then A' is \[n \times m\].
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Standard notation: A' or \[A^T\].
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Key properties: (A')' = A, (kA)' = kA', (A + B)' = A' + B', (AB)' = B'A'.
Concepts [12]
- Expansion of Determinant
- Minors and Co-factors
- Properties of Determinants
- Application of Determinants to Coordinate Geometry
- Methods of Solving Linear Equations in Two Variables > Determinant method (Cramer’s Rule)
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Concept of Matrices
- Types of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Matrix Multiplication
- Transpose of a Matrix
