Examples of inverse matrix in the following topics:
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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 $[B]$.
- Then calculate $[A^{-1}][B]$, that is, the inverse of matrix $[A]$, multiplied by matrix $[B]$.
- Practice using inverse matrices to solve a system of linear equations
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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.
- The inverse of matrix $[A]$, designated as $[A]^{-1}$, is defined by the property:
- The definition of an inverse matrix specifies that it must work both ways.
- In some cases, the inverse of a square matrix does not exist.
- Practice finding the inverse of a matrix and describe its properties
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- 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.
- Thus, to solve a system $AX=B$, for $X$, multiply both sides by the inverse of $A$ and we shall obtain the solution:
- Provided the inverse $\left( A^{-1} \right)$ exists, this formula will solve the system.
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- 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, $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$,
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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.
- Otherwise, it is equal to the additive inverse of its minor: $C_{ij}=-M_{ij}$
- Since $i+j=5 $ is an odd number, the cofactor is the additive inverse of its minor: $-(13)=-13$
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- 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.
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- Row multiplication (scale): Multiply a row of a matrix by a nonzero constant.
- Row addition (pivot): Add to one row of a matrix some multiple of another row.
- Since the matrix is essentially the coefficients and constants of a linear system, the three row operations preserve the matrix.
- For example, swapping two rows simply means switching their position within the matrix.
- Also, when solving a system of linear equations by the elimination method, row multiplication would be the same as multiplying the whole equation by a number to obtain additive inverses so that a variable cancels.
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- Recognize whether a function has an inverse by using the horizontal line test
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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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- Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
- Below is the graph of the parabola and its "inverse."
- Notice that the parabola does not have a "true" inverse because the original function fails the horizontal line test and must have a restricted domain to have an inverse.
- Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
- The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.