Examples of square matrix in the following topics:
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- The determinant of a $2\times 2$ square matrix is a mathematical construct used in problem solving that is found by a special formula.
- 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, $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$,
- Explain what a determinant represents, how to find one, and why only square matrices have them
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- The matrix that has this property is referred to as the identity matrix.
- 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 $0$:
- There is no identity for a non-square matrix because of the requirement of matrices being commutative.
- For a non-square matrix $[A]
$ one might be able to find a matrix $[I]$ such that $[A][I]=[A]$, however, if the order is reversed then an illegal multiplication will be left.
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- Cramer's Rule uses determinants to solve for a solution to the equation $Ax=b$, when $A$ is a square matrix.
- Cramer's Rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, i.e. a square matrix, valid whenever the system has a unique solution.
- It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.
- $\displaystyle
\left\{\begin{matrix} 3x+2y & = 10\\ -6x+4y & = 4 \end{matrix}\right.$
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- 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.
- Upper triangle form: A square matrix is called upper triangular if all the entries below the main diagonal are zero.
- 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 put a matrix in simplified form
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- The cofactor of an entry $(i,j)$ of a matrix $A$ is the signed minor of that matrix.
- In linear algebra, the cofactor (sometimes called adjunct) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices.
- 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.
- 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.
- Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors.
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- 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.
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- 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.
- Note also that only square matrices can have an inverse.
- 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 $[I]$.
- In some cases, the inverse of a square matrix does not exist.
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- 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.
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- 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.
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- 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 $[B]$.