matrix

Examples of matrix in the following topics:

• What is a Matrix?

  • The matrix has a long history of application in solving linear equations.
  • A matrix with m rows and n columns is called an m × n matrix or $m$-by-$n$ matrix, while m and n are called its dimensions.
  • A matrix which has the same number of rows and columns is called a square matrix.
  • In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
  • Each element of a matrix is often denoted by a variable with two subscripts.

• Matrix Structure

  • The matrix structure organizes employees by function and output to capitalize on strengths and improve efficiency.
  • The matrix structure groups employees by both function and product .
  • Balanced or functional matrix: A project manager is assigned to oversee the project.
  • Strong or project matrix: A project manager is primarily responsible for the project.
  • Representing matrix organizations visually has challenged managers ever since the matrix management structure was invented.

• Matrix Multiplication

  • When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.
  • If $A$ is an $n\times m $ matrix and $B$ is an $m \times p$ matrix, the result $AB$ of their multiplication is an $n \times p$ matrix defined only if the number of columns $m$ in $A$ is equal to the number of rows $m$ in $B$.  
  • Scalar multiplication is simply multiplying a value through all the elements of a matrix, whereas matrix multiplication is multiplying every element of each row of the first matrix times every element of each column in the second matrix.  
  • When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.
  • Each entry of the resultant matrix is computed one at a time.

• The Identity Matrix

  • The identity matrix $[I]$ is defined so that $[A][I]=[I][A]=[A]$, i.e. it is the matrix version of multiplying a number by one.
  • The matrix that has this property is referred to as the identity matrix.
  • The identity matrix, designated as $[I]$, is defined by the property:
  • What matrix has this property?
  • For a $3 \times 3$ matrix, the identity matrix is a $3 \times 3$ matrix with diagonal $1$s and the rest equal to $0$:

• Matrix Equations

  • It is possible to solve this system using the elimination or substitution method, but it is also possible to do it with a matrix operation.
  • Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: $X$ is the matrix representing the variables of the system, and $B$ is the matrix representing the constants.
  • Using matrix multiplication, we may define a system of equations with the same number of equations as variables as:
  • To solve a system of linear equations using an inverse matrix, let $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ be the constant matrix.
  • If the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.

• Determinants of 2-by-2 Square Matrices

  • It can be proven that any matrix has a unique inverse if its determinant is nonzero.
  • The determinant of a matrix $[A]$ is denoted $\det(A)$, $\det\ A$, or $\left | A \right |$.
  • In the case where the matrix entries are written out in full, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the brackets or parentheses of the matrix.
  • In linear algebra, the determinant is a value associated with a square matrix.
  • For a $2 \times 2$ matrix, $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$,

• Solving Systems of Equations Using Matrix Inverses

  • A system of equations can be readily solved using the concept of the inverse matrix and matrix multiplication.
  • A system of equations can be readily solved using the concepts of the inverse matrix and matrix multiplication.  
  • This can be done by hand, finding the inverse matrix of $[A]$, then performing the appropriate matrix multiplication with $[B]$.
  • Using the matrix function on the calculator, first enter both matrices.  
  • Then calculate $[A^{-1}][B]$, that is, the inverse of matrix $[A]$, multiplied by matrix $[B]$.

• Matrix Structure

  • The matrix structure is a type of organizational structure in which individuals are grouped via two operational frames.
  • In matrix management, the organization is grouped by any two perspectives the company deems most appropriate.
  • Blurred authority in a matrix structure can result in reduced agility in decision making and conflict resolution.
  • Matrix structures should generally only be used when the operational complexity of the organization demands it.
  • In a matrix structure, the organization is grouped by both product and function.

• Cofactors, Minors, and Further Determinants

  • The cofactor of an entry $(i,j)$ of a matrix $A$ is the signed minor of that matrix.
  • Specifically the cofactor of the $(i,j)$ entry of a matrix, also known as the $(i,j)$ cofactor of that matrix, is the signed minor of that entry.
  • The cofactor of $a_{ij}$ entry of a matrix is defined as:
  • In linear algebra, a minor of a matrix $A$ is the determinant of some smaller square matrix, cut down from $A$ by removing one or more of its rows or columns.
  • The determinant of any matrix can be found using its signed minors.

• A Geometrical Picture

  • Any vector in the null space of a matrix, must be orthogonal to all the rows (since each component of the matrix dotted into the vector is zero).
  • Similarly, vectors in the left null space of a matrix are orthogonal to all the columns of this matrix.
  • This means that the left null space of a matrix is the orthogonal complement of the column $\mathbf{R}^{n}$ .