Topic 1. Review of Matrices and Matrix Operations

• Matrix is a set of numbers or symbols arranged in a square or rectangular array of m rows and n columns.
• The size or dimension of a matrix is specified by its number of rows m and its number of columns n.
• Every number in the rows and columns of a matrix are referred to as elements.
• In a column matrix, the number of rows can be any positive integer, but the number of columns is one.
• Column matrix, also known as a vector, is a special type of matrix.
• In a row matrix, the number of columns can be any positive integer, but the number of rows is one.
• In a rectangular matrix, the number of rows and columns are m and n, respectively, where m and n are any positive integers.
• In a square matrix, the number of rows equals the number of columns.
• A square matrix is singular if its determinant is zero.
• If the determinant of a matrix is nonzero, it is termed nonsingular.
• In a symmetric matrix, the matrix is mirrored about the main diagonal going from top left to bottom right, it is always a square matrix.
• In a diagonal matrix, only the elements on the main diagonal are not zero and it is always a square matrix.
• Unit or “Identity” matrix is a diagonal matrix with 1’s along the main diagonal and usually identified by the symbol I.
• Matrices can be added or subtracted systematically as long as they have the same dimensions.
• If two matrices have equal dimensions, they are said to be conformable for addition or subtraction.
• Matrices that are conformable for addition or subtraction are true in the Commutative Law and Associative Law.
• In scalar multiplication, matrices can be multiplied by a scalar or known as a constant.
• If AB =0, neither A nor B are necessarily = 0.
• If AB = AC, B does not necessarily = C.
• The transpose of a matrix is a new matrix whose rows are the columns of the original matrix.
• The determinant of a matrix is a special number that can be computed from the elements of a square matrix.
• Geometrically, the determinant of a matrix can be viewed as the volume scaling factor of the linear transformation described by the matrix.
• The determinant of a matrix tells us things about the matrix that are useful in systems of linear equations, and it helps us find the inverse of a matrix.
• The symbol for determinant is single or double vertical lines both sides or simply det.
• The determinant obtained through the elimination of some rows and columns in a square matrix is called a minor of that matrix.
• The “cofactor” of a matrix is a “signed” version of a minor of a matrix.
• The determinant of a mxn matrix is equal to the sum of the products of the elements of any one row or column and their cofactors.
• The Adjoint of a matrix is the transpose of its cofactor matrix.
• When A is multiplied by A-1 the result is the identity matrix I.
• Non-square matrices do not have inverses.
• A square matrix has an inverse if and only if the determinant of that matrix is not zero.
• Inverse of a matrix is determined by adjoint method and by elementary row transformations (i.e., Gauss-Jordan Elimination method).
• In linear equations by matrix method, A is the matrix of coefficients, X is the matrix of unknowns, and B is the matrix of known values.