All About Matrices

Matrix Order

• The order of a matrix with 3 rows and 4 columns is 3 × 4.
• A square matrix has an equal number of rows and columns.

Types of Matrices

• A matrix with only one row is a row matrix.
• A scalar matrix has equal diagonal elements and non-diagonal elements as zero; e.g., [5 0; 0 5].
• A diagonal matrix has all non-diagonal elements as zero.

Matrix Operations

• Matrix addition is defined only when matrices have the same order.
• For matrices A and B of the same order, (A + B)^T = A^T + B^T.
• Scalar multiplication means multiplying a matrix by a scalar value.
• Matrix multiplication is associative and distributive.
• 4×3 × 2×3 is not a valid matrix multiplication because the number of columns in the first matrix does not equal the number of rows in the second matrix.

Transpose of a Matrix

• The transpose of a transpose is the original matrix.
• If matrix A is symmetric, then A^T = A.
• If matrix A is skew-symmetric, then A^T = -A.

Identity Matrix

• The identity matrix I satisfies AI = A and IA = A.

Zero Matrix

• For any matrix A, A - A = 0.

Determinants

• The determinant is defined only for square matrices.
• The value of the determinant of the identity matrix of order 3 is 1.
• If two rows of a determinant are equal, then its value is 0.
• |AB| = |A| × |B|.
• |A^T| = |A|.

Elementary Row Operations

• Elementary row operations do not include adding a scalar to a matrix.
• Multiplying a row by another row is not a valid matrix row operation.
• Elementary operations are used to find the inverse of a matrix.
• When two rows are interchanged, the determinant changes sign.
• If a row is multiplied by a scalar k, the determinant is multiplied by k.

Echelon Forms

• A matrix is in row echelon form if each leading entry is 1 and to the right of the leading entry in the row above, and zero rows (if any) are at the bottom.
• In reduced row echelon form, every leading 1 is the only non-zero entry in its column.
• The reduced echelon form is unique for any matrix.
• Row echelon form is helpful in solving systems of equations.

Rank of a Matrix

• The rank of a matrix is the number of non-zero rows in echelon form.
• The rank of a zero matrix is 0.
• The maximum rank of an m × n matrix is min(m, n).
• If the rank of a matrix is equal to the number of unknowns, the system has a unique solution.
• Rank remains unchanged under transposition, elementary row operations, and multiplication with a scalar.

Invertible Matrices

• A matrix A is invertible only if it is square, |A| ≠ 0, and Rank(A) = order of A.
• If a matrix A is such that A^2 = A, then A is idempotent.

Solving Linear Systems

• In solving AX = B, if A is non-singular, the solution is X = A⁻¹B.
• The inverse of a matrix exists only if the determinant ≠ 0.
• A matrix is orthogonal if A^T A = I.
• It is false that every square matrix has an inverse.
• The determinant of a product equals the product of the determinants.
• Elementary row operations do not change the rank of a matrix.
• It is false that the rank of a matrix is equal to the number of non-zero columns.
• Reduced row echelon form cannot have non-zero entries above and below pivots.
• The determinant of a triangular matrix is the product of its diagonal elements.
• The rank of a matrix gives the maximum number of linearly independent rows or columns.
• The matrix A is called symmetric if A^T = A.
• In echelon form, each leading entry is to the right of the leading entry in the row above.
• A matrix with all elements zero is called a null matrix.

Systems of Linear Equations

• A system of linear equations has a solution if it is consistent.
• A system of equations is inconsistent if it has no solution.
• A system is consistent and has infinite solutions when Rank < number of variables but equals rank of augmented matrix.
• A system is uniquely solvable if the Rank of the coefficient matrix = rank of augmented matrix = the number of variables.
• A system of 3 equations in 3 variables is consistent and has a unique solution if the determinant is non-zero.
• A homogeneous system always has a trivial solution.
• Homogeneous systems have non-trivial solutions when the determinant = 0.
• A system is non-homogeneous if at least one RHS value ≠ 0.
• A homogeneous system with determinant = 0 may have infinite solutions.
• The trivial solution means all variables are zero.

Gauss Elimination

• The Gauss Elimination method converts the matrix into row echelon form.
• The final goal of Gauss-Jordan elimination is to reach reduced row echelon form.
• In Gauss-Jordan method, leading entries in pivot columns are ones.
• Gauss Elimination is used to solve a system of linear equations.
• In Gauss Elimination, back-substitution starts from the last row.

Free Variables

• Free variables exist when the number of unknowns > rank.
• Free variables can take any value.
• Free variables indicate infinite solutions exist.
• In RREF, a column with no pivot corresponds to a free variable.
• Free variables are found in both homogeneous and non-homogeneous systems.

Types of Systems

• A system with one solution is called consistent and independent.
• A system with infinite solutions is consistent and dependent.
• A system with no solution is called inconsistent.
• If the augmented matrix has a row like [0 0 0 | 5], the system is inconsistent.
• If rank(A) ≠ rank(A|B), the system is inconsistent.
• AX = B is called the matrix form of a linear system.
• In matrix AX = 0, the system is homogeneous.
• The augmented matrix includes coefficients and constants.
• Matrix form AX = B can be solved using row operations.
• If rank(A) = rank(A|B) < the number of variables, then the system has infinite solutions.

True/False Statements

• Every homogeneous system is consistent.
• A free variable increases the number of solutions.
• If a system has a unique solution, it must have no free variables.
• Gauss elimination method does not always give a unique solution.
• A system with 2 equations and 3 variables cannot have a unique solution.
• Gauss-Jordan elimination can help find the inverse of a matrix.
• A pivot is the leading non-zero entry in a row.
• If all entries in a row are zero, that row is ignored in rank computation.
• It is false that a system with more variables than equations always has infinite solutions.
• Inconsistent systems arise when equations contradict each other.

Vector Spaces

• The set of all real-valued polynomials is a vector space over ℝ.
• A set V is a vector space over field F if it satisfies all vector space axioms.
• The set of all 2×2 matrices with determinant 1 is not a vector space.
• The set of all vectors in ℝ³ forms a vector space because it is closed under addition and scalar multiplication.
• The zero vector in a vector space is always unique.

Subspaces

• A subset W of a vector space V is a subspace if W is closed under addition and scalar multiplication.
• The set {0} is a subspace of every vector space.
• The set of all vectors with positive components is not a subspace of ℝ².
• The set of all symmetric 2×2 matrices is a subspace of M₂×₂ (2×2 matrices).
• If W is a subspace of V, then 0 ∈ W.

Linear Combinations

• A vector v is a linear combination of vectors v₁, v₂,..., vn if v = a₁v₁ + a₂v₂ +... + anvn for some scalars a₁,..., an.
• The zero vector is a linear combination of any set of vectors because all scalar coefficients can be zero.
• The expression 3v₁ + 0v₂ − 2v₃ is an example of a linear combination.
• If a vector v cannot be written as a linear combination of other vectors in a set, then it is linearly independent of that set.
• (2, 3) is a linear combination of vectors (1,0) and (0,1).

Span

• The span of vectors {(1, 0), (0, 1)} in ℝ² is all of ℝ².
• A set of vectors spans a space if every vector in the space is a linear combination of the set.
• The set {(1, 0, 0), (0, 1, 0), (0, 0, 1)} spans ℝ³.
• Span{(1, 2)} in ℝ² is a line through the origin.
• If a vector lies in the span of a set, it can be written as a linear combination of vectors in the set.
• Every vector space contains infinitely many subspaces.
• The set of all 2x2 matrices forms a vector space.
• A subspace is always a vector space.
• A vector space must contain the zero vector.
• It is false that a spanning set of ℝ² must contain exactly two vectors.
• A if span{v₁, v₂} = ℝ², then v₁ and v₂ are linearly independent.
• A vector space cannot be both infinite and finite-dimensional.
• Not just any subset of a vector space is a subspace.

True/False statements

• If a vector space is 3-dimensional, any set of 4 vectors must be linearly dependent.
• A single non-zero vector can span a line in ℝ².
• The set of all continuous real-valued functions is a valid example of a vector space.
• All elements do not have to be positive for a set to be a vector space.
• The set of all 2×2 diagonal matrices is a subspace of M₂×₂.
• All vectors of the form (x, 0, z) is a subspace of ℝ³.
• A Subspace must contain the zero vector
• A set of vectors {v₁, v₂,..., vn} is linearly dependent if at least one vector is a linear combination of the others
• (1,3) is not a linear combination of (1, 2)
• If v = a₁v₁ + a₂v₂ +... + anvn, where aᵢ are scalars, then v is a linear combination of v₁, v₂,..., vn
• 2(1, 2, 3) + 3(0, 1, −1) is a linear combination
• In ℝ², the vector (5, −1) is a linear combination of (1, 0) and (0, 1) because: (5, −1) = 5(1, 0) + (−1)(0, 1)
• If a set spans ℝ², it must contain at least wo linearly independent vectors
• The span of the zero vector is: {0}
• The vectors (1, 2) and (2, 4) do not span ℝ² because they are linearly dependent
• The span of {(1, 2), (3, 6)} is a line in ℝ²
• The span of a single non-zero vector in ℝ³ is a line through the origin
• Every subspace of a vector space is itself a vector space.
• A line that passes through the origin in ℝ² is a subspace.
• A set of vectors that spans a vector space must be linearly dependent is false
• The number of vectors in a spanning set is always equal to the dimension. It is false.
• Linear combinations form the basis of defining span.