This chapter introduces matrices, which are essential tools in various fields of mathematics and science. Understanding matrices helps simplify complex mathematical operations and solve systems of linear equations.
Matrices – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - I, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Matrices chapter of Mathematics Part - I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
A = [a_ij], m × n
A denotes a matrix with m rows and n columns. It is structured as a rectangular array of elements a_ij.
A' = [a_ji], n × m
The transpose of matrix A is formed by interchanging its rows and columns, resulting in a matrix of order n × m.
A + B = [a_ij + b_ij]
The addition of matrices A and B is performed element-wise, where A and B must have the same order.
kA = [k * a_ij]
Multiplying matrix A by a scalar k scales each element by k.
A - B = A + (-B)
The difference of two matrices is defined as the sum of the first matrix and the negative of the second matrix.
AB = C, (where C is defined)
Matrix multiplication AB is defined if the number of columns in A equals the number of rows in B.
(AB)' = B'A'
The transpose of the product of two matrices A and B equals the product of the transposes in reverse order.
A + A' is symmetric
The sum of a matrix and its transpose results in a symmetric matrix.
A - A' is skew symmetric
The difference between a matrix and its transpose results in a skew symmetric matrix.
A^(-1) exists if AB = BA = I
Matrix A is invertible if there exists a matrix B such that their product results in the identity matrix.
Equations
A + B = [a_ij + b_ij]
Addition of two matrices, element by element, where A and B must be of the same dimension.
A - B = [a_ij - b_ij]
Element-wise subtraction of matrices A and B, defined only if A and B share the same order.
kA = [ka_ij]
Scalar multiplication of matrix A by k multiplies each element of A by k.
A' = [a_ji]
The transpose of A is obtained by swapping its rows and columns.
AB = C (with dimensions m × p)
Matrix multiplication is valid when the number of columns in A equals the number of rows in B.
det(A) = a_11 * C_11 - a_12 * C_12 + ... + (-1)^{1+j} * a_1j * C_1j
Formula for computing the determinant of a matrix A using cofactor expansion.
(AB)C = A(BC)
Matrix multiplication is associative; the order of operations does not change the result.
A + B = B + A
Matrix addition is commutative; the order of summands does not affect the outcome.
A^(-1)A = I
The inverse of A multiplied by A returns the identity matrix.
rank(A) ≤ min(m, n)
The rank of a matrix A cannot exceed the smaller of the number of its rows or columns.
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