Here you will learn what is derivative or differentiation and various differentiation formulas class 12. Let’s begin – What is Derivative or Differentiation ? Let f(x) be a differentiable or derivable function on [a, b]. Then, \(lim_{h \to 0}\) \(f(x + h) – f(x)\over h\) or, \(lim_{h \to 0}\) \(f(x – h) – f(x)\over -h\) …
Here you will learn how to find the determinant of matrix 2×2 with examples. Let’s begin – Determinant of Matrix 2×2 If A = \(\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\) is a square matrix of 2×2, then \(a_{11}a_{22} – a_{12}a_{21}\) is called the determinant of A. i.e. | A | = \(\begin{vmatrix} …
Here you will learn what is determinant of matrix and formula for how to find the determinant of matrix of different order. Let’s begin – What is Determinant ? If the equations \(a_1x + b_1\) = 0, \(a_2x + b_2\) = 0 are satisfied by the same value of x, then \(a_1b_2 – a_2b_1\) = …
Here you will learn how to find minors and cofactors of a matrix of order 3×3 and 2×2 with examples. Let’s begin – How to find Minors and Cofactors of a Matrix Minor of Matrix (3×3 and 2×2) Let A = \([a_{ij}]\) be a square matrix of order n. The minor \(M_{ij}\) of \(a_{ij}\) in …
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Here you will learn properties of determinant of matrix with examples. Let’s begin – Properties of Determinant of Matrix Property 1 : The value of determinant remains unaltered or unchanged, if the rows & columns are inter-changed, e.g. if D = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 …
Her you will learn how to find determinants of matrix 4×4 with example. Let’s begin – Determinants of Matrix 4×4 To evaluate the determinant of a square matrix of order 4 we follow the same procedure as discussed in previous post in evaluating the determinant of a square matrix of order 3. If A = …
Here you will learn how to find determinants of matrix 3×3 with examples. Let’s begin – Determinants of Matrix 3×3 If A = \(\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\) is a square matrix of order 3, then \(a_{11}(a_{22}a_{33} – a_{23}a_{a32})\) – \(a_{12}(a_{33}a_{21} …
Here you will learn formula for inverse of a matrix and properties of inverse of matrix with example. Let’s begin – Formula for Inverse of a Matrix A square matrix A said to be invertible if and only if it is non-singular (i.e. |A| \(\ne\) 0) and there exists a matrix B such that, AB …
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Here you will learn how to find adjoint of the matrix 2×2 and 3×3, cofactors and its properties with examples. Let’s begin – Adjoint of the Matrix Let A = \([a_{ij}]\) be a square matrix of order n and let \(C_{ij}\) be a cofactor of \(a_{ij}\) in A. Then the transpose of the matrix of …
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Here you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example. Let’s begin – Trace of Matrix The sum of the elements of the square matrix A lying along the principal diagonal is called the trace of A i.e (tr(A)). Thus if A = \([a_{ij}]_{n\times n}\), then …
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