empty matrix

Examples of empty 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.

• 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 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.

• 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 $[B]$.

• 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.

• Simplifying Matrices With Row Operations

  • Augmented matrix: an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
  • A triangular matrix is one that is either lower triangular or upper triangular.
  • A matrix that is both upper and lower triangular is a diagonal matrix.
  • Use elementary row operations to reduce the matrix to reduced row echelon form:
  • Use elementary row operations to put a matrix in simplified form

• The Inverse of a Matrix

  • The matrix $B$ is the inverse of the matrix $A$ if when multiplied together, $A\cdot B$ or $B\cdot A$ gives the identity matrix.
  • The definition of an inverse matrix is based on the identity matrix $[I]$, and it has already been established that only square matrices have an associated identity matrix.
  • When multiplying this mystery matrix by our original matrix, the result is $[I]$.
  • If an inverse has been found, then a quick check to be sure it is correct is to multiply it by the original matrix and see if the identify matrix results:
  • This is called a singular matrix.

• Addition and Subtraction; Scalar Multiplication

  • Matrix addition is commutative and is also associative, so the following is true:
  • Just add each element in the first matrix to the corresponding element in the second matrix.
  • Note that element  in the first matrix, $1$, adds to element $x_{11}$ in the second matrix, $10$, to produce element $x_{11}$ in the resultant matrix, $11$.
  • Multiplying a matrix by $3$ means the same thing; you add the matrix to itself $3$ times, or simply multiply each element by that constant.
  • The resulting matrix has the same dimensions as the original.