Matrices and Determinants

Definitions

$An m \times n matrix A is a rectangular$

$array of elements with m rows and n$

$colums$

$A = [a_{i j}] = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{z z} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}]$

Square Matrix

$Square matrix is of order n \times n .$

$A square matrix [a_{i j}] is symmtrc if a_{i j} = a_{j i .}$

$A square matrix [a_{i j}] is skew - symmtrc if a_{i j} = - a_{j i .}$

Diagonal Matrix

$Diagonal matrix is a square matrix with all$

$elements zero except those on the leading$

$diagonal .$

Unit Matrix

$Unit matrix is a diagonal matrix in which$

$all elements on the leading diagonal$

$are 1 . Unit matrix is denoted by I .$

Null Matrix

$A null matrix is one whose all elements are 0 .$

Operations with Matrics

Addition/Subtraction

$Two matrices A and B are equal if and$

$only if they are both the same shape and$

$corresponding elements are equal .$

$Two matrices can be added (or subtracted)$

$if and only if they have the same shape m \times n .$

$A = [a_{i j}] = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}]$

$B = [b_{i j}] = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{22} & \dots & b_{2 n} \\ ⋮ & ⋮ & ⋮ \\ b_{m 1} & b_{m 2} & \dots & b_{m n} \end{matrix}]$

$A + B = [a_{i j} + b_{i j}] = [\begin{matrix} a_{11} + b_{11} & a_{12} + b_{12} & \dots & a_{1 n} + b_{1 n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \dots & a_{2 n} + b_{2 n} \\ ⋮ & ⋮ & ⋮ \\ a_{m 1} + b_{m 1} & a_{m 2} + b_{m 2} & \dots & a_{m n} + b_{m n} \end{matrix}]$

Scaler Multiplication

$If k is a scaler, and A is a matrix, then$

$k A = [{k a}_{i j}] = [\begin{matrix} {k a}_{11} & {k a}_{12} & \dots & {k a}_{1 n} \\ {k a}_{21} & {k a}_{22} & \dots & {k a}_{2 n} \\ ⋮ & ⋮ & ⋮ \\ {k a}_{m 1} & {k a}_{m 2} & \dots & {k a}_{m n} \end{matrix}]$

Multiplication

$Two matrices can be multiplied together$

$only when number of colums in the first$

$is equal to number of rows in the second .$

$If$

$A = [a_{i j}] = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}]$

$and$

$B = [b_{i j}] = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1 k} \\ b_{21} & b_{22} & \dots & b_{2 k} \\ ⋮ & ⋮ & ⋮ \\ b_{n 1} & b_{n 2} & \dots & b_{n k} \end{matrix}]$

$C = A B = [c_{i j}] where$

$c_{i j} = \underset{λ = 1}{\sum^{n}} a_{i λ} b_{λ j}$

$If m \times n matrix is multiplied with n \times k$

$matrix then result is a m \times k matrix .$

Transpose of a Matrix

$If the rows and columns of a matrix are$

$interchanged then the new matrix is$

$called the transpose of the original matrix .$

$If A is the original matrix, its transpose$

$is denoted A^{T} .$

$If {A A}^{T} = I then A is orthogonal matrix .$

$If A B is defined, then$

${(A B)}^{T} = B^{T} A^{T}$

Positive Integral Powers

$If A is an n \times n matrix, then we define$

$A^{2} = A A, A^{3} = (A A A) and so on .$

$In general, A^{n} = (A A \dots n times) .$

$Also, A^{0} = I_{n} where I_{n} is an identity$

$matrix of order n .$

Matrix Polynomial

$Let f (x) = \underset{i = 0}{\sum^{m}} a_{i} x^{i} be a polynomial of$

$degree m, If A is a square matrix of$

$order n . Then we define$

$f (A) = \underset{i = 0}{\sum^{m}} a_{i} A^{i}$

Properties of Matrix Operations

$1 . Matrix addition is commutative .$

$A + B = B + A$

$2 . Matrix addition is associative .$

$(A + B) + C = A + (B + C)$

$3 . If A is an m \times n matrix and O is a m \times n null$

$matrix, then$

$A + O = O + A = A$

$4 . Negative of a matrix A = {[a_{i j}]}_{m \times n}$

$- A = {[- a_{i j}]}_{m \times n}$

$5 . Subtration of two matrices$

$A - B = A + (- B)$

$6 . A - A = A + (- A) = (- A) + A = O$

$7 . k (A + B) = k A + k B, where k scaler$

$8 . (k_{1} + k_{2}) A = k_{1} A + k_{2} A, where k_{1}, k_{2} scaler$

$9 . k_{1} (k_{2} A) = (k_{1} k_{2}) A, where k_{1}, k_{2} scaler$

$10 . {(A^{T})}^{T} = A$

$11 . {(A + B)}^{T} = A^{T} + B^{T}$

$12 . {(k A)}^{T} = k A^{T}, k scaler$

$13 . Symmetric matrix A^{T} = A$

$14 . Skew - symmetric matrix A^{T} = - A$

$15 . Every diagonal element of$

$skew - symmetric matrix is 0 .$

$16 . The sum of two symmetric matrix is symmetric .$

$17 . The sum of two skew - symmetric matrix is skew - symmetric .$

$18 . For any square matrix$

$(A + A^{T}) is symmetric$

$(A - A^{T}) is skew - symmetric$

$19 . Matrix multiplication is not$

$commutative is general .$

$20 . Matrix multiplication is associative .$

$(A B) C = A (B C)$

$21 . Multiplication distributes addition .$

$A (B + C) = A B + A C$

$(A + B) C = A C + B C$

$22 . If A is an m \times n matrix$

${A I}_{n} = A$

$I_{m} A = A$

$wherw I_{n} identity matrix of order n .$

$23 . If A is a matrix and O is null matrix,$

$A_{m \times n} O_{n \times p} = O_{m \times p}$

$O_{p \times m} A_{m \times n} = O_{p \times n}$

$24 . If A and B are two matrices such that$

$A B is defined then$

${(A B)}^{T} = B^{T} A^{T}$

$25 . If A and B are two matrices such that$

$A B is defined then$

$A (- B) = - (A B)$

$(- A) (B) = - (A B)$

$26 . A (B - C) = A B - A C$

Determinants

$Corresponding to each square matrix$

$[\begin{matrix} a_{11} & \dots & a_{1 n} \\ ⋮ & ⋱ & ⋮ \\ a_{n 1} & \dots & a_{n n} \end{matrix}]$

$A = [a_{i j}] there is associated an expression$

$called the d e t e r m i n a n t o f A denoted by$

$\det A or ∣ A ∣, written as$

$\det A =∣ A ∣= | \begin{matrix} a_{11} & \dots & a_{1 n} \\ ⋮ & ⋱ & ⋮ \\ a_{n 1} & \dots & a_{n n} \end{matrix} |$

Value of a Determinant

$Value of determinant of order 2$

$| \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} | = (a_{11} a_{22} - a_{12} a_{21})$

$Minor of a_{i j} in ∣ A ∣$

$The minor of an element a_{i j} in ∣ A ∣ is defined$

$as the value of determinant obtained by$

$deleting i th row and j th column of ∣ A ∣,$

$and is denoted by M_{i j} .$

$Co - factor of a_{i j} in ∣ A ∣$

$The co - factor C_{i j} of an element a_{i j} is$

$defined as C_{i j} = {(- 1)}^{i + j} \cdot M_{i j}$

$Value of a determinant$

$The value of a determinant is the sum of$

$the products of element of a row (or column)$

$with their corresponding co - factors .$

$A determinant may be expanded by$

$arbitarily chosen row or column .$

$Expansion of a determinant or order n$

$Expansion by i th row$

$\det A = \underset{j = 1}{\sum^{n}} a_{i j} C_{i j}$

$Expansion by j th column$

$\det A = \underset{i = 1}{\sum^{n}} a_{i j} C_{i j}$

Properties of Determinants

$1 . The value of a determinant remains$

$unchanged if its row and columns are$

$interchanged .$

$2 . If two rows or columns of a determinant are$

$interchanged, the sign of determinant$

$is changed but the absolute value$

$remains same .$

$3 . If two rows (or two columns) are$

$identical, the value of the determinant$

$is 0 .$

$4 . If the element of any row or column$

$are multiplied by a common factor,$

$the determinant is multiplied by$

$that factor .$

$5 . If the elements of any row (or column)$

$are increased (or decreased) by equal$

$multiples or corresponding elements of$

$any other row (or column), the value$

$of the determinant is unchanged .$

Adjoint of a Matrix

$If A is a square n \times n matrix, its adjoint,$

$denoted by adj A, is the transpose of the$

$matrix of the matrix if cofactors C_{i j} of A .$

$adj A = {[C_{i j}]}^{T}$

Inverse of a Matrix

$If A is a square matrix with a nonsingular$

$∣ A ∣ then its inverse A^{- 1} is given by :$

$A^{- 1} = \frac{adj A}{\det A}$

$If matrix product A B is defined then$

${(A B)}^{- 1} = B^{- 1} A^{- 1}$

Invertible Matrix

$1 . {A A}^{- 1} = A^{- 1} A = I$

$2 . A B = A C \Rightarrow B = C, if ∣ A ∣\neq 0$

$3 . {(A B)}^{- 1} = B^{- 1} A^{- 1}, if ∣ A ∣, ∣ B ∣\neq 0$

$4 . {(A^{T})}^{- 1} = {(A^{- 1})}^{T}, if ∣ A ∣\neq 0, if ∣ A ∣\neq 0$

$5 . adj (A B) = (adj B) (adj A), if ∣ A ∣, ∣ B ∣\neq 0$

$6 . {(adj A)}^{T} = adj A^{T}, if ∣ A ∣\neq 0$

$7 . ∣ adj A ∣=∣ A ∣^{n - 1}, if ∣ A ∣\neq 0$

$8 . adj (adj A) =∣ A ∣^{n - 2} A$

Elementary Row and Column Transformation of a Matrix

The following are three elementary transformation of a matrix $R_{n}$ indicates $n t h$ row.

Interchange of any Two Rows or Two Columns ( $R_{m} = R_{n}, R_{n} = R_{m}$ or $C_{m} = C_{n}, C_{n} = C_{m}$ )
Multiplication of Row or Column by a Non-zero Number ( $R_{n} = k R_{n}$ or $C_{n} = k C_{n}$ )
Multiplication of Row or Column by a Non-zero Number and Add the Result to the Other Row or Column ( $R_{n} = R_{n} + k R_{m}$ or $C_{n} = C_{n} + k C_{m}$ )

Linear Independence

Row $R_{1}, R_{2}, \dots, R_{n}$ with same number of columns are linearly independent if

$a_{1} R_{1} + a_{2} R_{2} + \dots + a_{n} R_{n} = R_{0}$ , implies that $a_{i} = 0$ for $i = 1, 2, \dots n$ , $R_{0}$ is row with all columns with values 0.

Columns $C_{1}, C_{2}, \dots, C_{n}$ with same number of rows are linearly independent if

$a_{1} C_{1} + a_{2} C_{2} + \dots + a_{n} C_{n} = C_{0}$ , implies that $a_{i} = 0$ for $i = 1, 2, \dots n$ , $C_{0}$ is column with all row with values 0.

Vectors $v_{1}, v_{2}, \dots, v_{n}$ with same number of dimensions are linearly independent if

$a_{1} v_{1} + a_{2} v_{2} + \dots + a_{n} v_{n} = 0$ , implies that $a_{i} = 0$ for $i = 1, 2, \dots n$

Rank of a Matrix

The maximum number of linearly independent columns (or rows) of a matrix is called the rank of a matrix. The rank of a matrix cannot exceed the number of its rows or columns.

Key Concepts:

Linearly Independent Rows/Columns: A set of vectors (rows or columns) are linearly independent if no vector in the set can be written as a linear combination of the others. The rank indicates how many of the rows or columns are linearly independent.
Row Rank and Column Rank: The row rank is the number of linearly independent rows, and the column rank is the number of linearly independent columns. Row rank is always equal to column rank for any matrix, which is why we just call it the rank of the matrix.

Methods to Find the Rank:

Row Echelon Form (REF):
- Transform the matrix into row echelon form (REF) using Gaussian elimination.
- The rank is the number of non-zero rows in the REF.
Reduced Row Echelon Form (RREF):
- Transform the matrix into reduced row echelon form (RREF), which is a further simplified version of REF.
- The rank is the number of non-zero rows in the RREF.
Determinant Method (for square matrices):
- For a square matrix, if the determinant is non-zero, the rank is equal to the number of rows (or columns). If the determinant is zero, the matrix has rank less than its size.
Singular Value Decomposition (SVD):
- For a matrix AAA, the rank can also be determined by the number of non-zero singular values in its Singular Value Decomposition (SVD).

Minor Method: If the rank of matrix A is r, then there exists at least one minor of order r which does not vanish. Every minor of matrix A of order (r + 1) and higher-order (if any) vanishes.

Row Echelon Form

Specifically, a matrix is in row echelon form if

All rows consisting of only zeroes are at the bottom
The leading entry (that is the left-most nonzero entry) of every nonzero row is to the right of the leading entry of every row above
To convert a matrix to Row Echelon Form (REF), you perform a series of Gaussian elimination steps. The goal is to make the matrix satisfy the following conditions:
All nonzero rows are above any rows of all zeros.
The leading entry (also called the pivot) in each nonzero row is 1.
The pivot in any row appears to the right of the pivot in the row above it.
All entries below a pivot are zero.

Step-by-Step Process to Covert to Row Echelon From

Identify the first non-zero element in the first column. This element becomes your pivot.
Swap rows (if necessary) to make sure the pivot is at the top of the column (i.e., the pivot should be the first non-zero entry in the row).
Scale the row (if necessary) to make the pivot equal to 1. This can be done by dividing the entire row by the value of the pivot element.
Eliminate entries below the pivot: Use row operations to create zeros below the pivot. This is done by subtracting multiples of the pivot row from the rows below it.
Move to the next column: Once the pivot in the first column is dealt with, move to the next column and repeat the process for the remaining submatrix (i.e., ignore the row and column where the pivot was placed).
Continue until all pivots are in place: Repeat the process until you have processed all columns.

Reduced Row Echelon Form

To convert a matrix to Reduced Row Echelon Form (RREF), you need to apply Gaussian-Jordan elimination. The goal is to satisfy the following conditions:

The first non-zero entry in each row (the pivot) is 1.
The pivot is the only non-zero entry in its column (i.e., all entries above and below the pivot are zero).
The pivots in each row move to the right as you move down the rows.
All rows of zeros are at the bottom of the matrix.

Problems in Matrices and Determinants

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