一个rrange the equations into the form:

, where a,b,c,d are constants.

Then we have the system of equations:.

The augmented matrix is found by copying the constants into the respective rows and columns of a matrix.

The vertical line in the matrix is analogous to the = sign thus resulting in the following:

Create an augmented matrix by entering the coefficients into one matrix and appending a vector to that matrix with the constants that the equations are equal to. Then you can row reduce to solve the system.

First, lets make this augmented matrix:

Now we can row reduce the matrix using the three row reduction operations: mutliply a row, add one row to another, swap row positions.

First, we can subtract.

Swap

Subtract

一个dd

Subtract

Mulitply

You can stop here given that this augmented matrix can be rewritten as a system again with

, or you can continue using the matrix, subtracting multiples offrom the other two rows to get an identity matrix yielding the solution to the system.

For two matrices to be equal, two conditions must hold:

1) The matrices must have the same dimension. This can be seen to be the case, since both matrices have two rows and one column.

2) All corresponding entries must be equal. For this to happen, it must hold that

This is a system of two equations in two variables, which can be solved as follows:

一个dd both sides of the equations:

It follows that

Substitute back:

Thus, there exists a solution to the system, and, consequently, to the original matrix equation. The statement is therefore false.

First, it must be established thatis defined. This is the case if and onlyandhave the same number of rows, which is true, and they have the same number of columns, which is also true.is therefore defined.

一个ddition of two matrices is performed by adding corresponding elements, so

This is the two-by-two identity. Therefore,.

传统上在减少简化阶梯形(第一行row's values are 1,0,0; second row's values are 0,1,0; third row's values are 0,0,1), x is depicted in the first row,in the second row, andin the third. Therefore the correct answer is.

The system of equations in matrix form is written:

Beginning with the top equation, insert each coefficient into the top row of the matrix, from left to right. Then, place the 2 on the right hand side of the vertical line (anytime you see this vertical line in a matrix, it is called anaugmentedmatrix.Because the first equation's coefficients are 1, 1, and 1, these form the top row of the matrix, with 2 on the right side of the bar. Then, move on to the second row, which has coefficients 2, -1, and 3. Place these into the second row of the matrix, with -7 on the right hand side of the vertical bar. Finally, take the coefficients from the 3rd equation, 1, 1, and -1, and place these in the bottom row with 6 on the right hand side of the vertical bar.

Thesystem of equations in matrix form is written: