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Solution Using Microsoft Math Solver
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USING DESMOS MATRIX CALCULATOR
Trace is the sum of numbers from top left moving diagonally to the bottom right of the matrix
Trace of a matrix is equal to the sum of its eigenvalues. This propoerty is useful in various areas of linear algebra and quantum mechanics.
If two matrices are similar, they have the same trace number. This helps in identiying matrices that represent the same linear transformation.
RREF button means Reduced Row Echelon Form. RREF of a given matrix is a simplified version of the matrix that makes it easier to solve linear equations. A matrix is in RREF if it meets the following criteria:
1. Leading 1s: Each leading entry (the first non-zero number from the left in a row ) is 1.
2. Zeros above and below: Each leading 1 is the only non-zero number in its column.
3. Row order: The leading 1 in each row is to the irght of the leading 1 in the row above it.
4. Zero rows: Any rows consisting entirely of zeros are at the bottom of the matrix.
Practice solving 3 x 3 matrix
Practice solving 4 x 4 matrix
Practice solving A * B matrix multiplication
Matrix A = 2 ; Matrix B = 3
LESSON NO.1 Matrix A = m x n size ; the letter m - tells you how many rows of number, this number is also called component, elements, or entries in many linear algebra book. You must understand matrix to solve simultaneous linear equations. In linear equations , m is the variable letter assigned to tell you how many counts or rows of equations. It is simultaneous equation because all the answer must simultaneously satisfy all the equation 1 , equation 2, equation 3 and so on. Equations are usually the conditions or constraints that must be satisfied.
The variable letter n - tells you how many columns of number or entries . In linear algebra , n is the variable letter assigned to tell you how many unknown variables are present in an equation. n-tuples means tell me how many unknown variables are present in one row of equation. This tuples terminology is used during the days of one of the great mathematician name Euclid. This tuples (letter - n ) can be seen in many mathematics textbook written as R n. The variable letter R means Root or Real number , the solution set that satisfy all the equations.
DEMONSTRATE THAT YOU UNDERSTAND LESSON NO.1 Now apply lesson 1
When you see Matrix A = 3 x 2 explain how many rows and column of numbers you will expect to see as your data input. Can you write an example of 3 x 2 matrix. With this knowledge and understanding you can tell if somebody is making a mistake in translating linear equations in matrix form.
LESSON NO. 2 What is determinant? Determinant is the answer after you have done the calculation of square matrix.
LESSON NO. 3 What is a square matrix? Square matrix means the number of row data and column data are equal. For example the above matrix is called 2 x 2 square matrix because if you count the number of row = 2 and if you count the number of column = 2. Hence , it is called 2 x 2 Square Matrix or Just 2 x 2 Matrix.
LESSON NO. 4 How do you compute the value of determinant? Calculation of determinant is time consuming as you can see from calculation shown below. First you need to have a calculator and you must remember from your memory the formula and then the procedure on how to do the computation.
Why do I need to acquire the knowledge and skill on how to compute for the determinant value?
Answer: By acquiring the knowledge and skill on how to compute for determinant value of 2 x 2 matrix, it will help you to use Excel software or other computer programming language to solve 2 simultaneous linear equation quickly.
Shown below is the time consuming manual calculation on how to solve two simultaneous linear equation using determinant and Cramer's Rule or Cramer's Procedure. Cramer's Procedure only works if the determinant value is not equal to zero. Why ? because When Determinant, D = 0; 1 / Determinant or 1 / 0 = ?? No Unique answer. We also say I can't determine the answer or indeterminant. In higher mathematics using approximation, we can still solve indeterminate problem like division by zero using approximation techniques. We assume the divisor is approaching the limit 0.
When you go to step 5 below, imagine your determinant D value = 0 not -2. So now you are going to divide -14 / 0 = ??? , you say division by zero is not possible. Indeterminant, you can't determine the answer.
The same explanation for inverse of matrix A, Matrix A has an inverse, A-1, only if the determinant value of matrix A is not equal to zero.
X = B A-1, Inverse of matrix A is important in solving linear equations. It can be written as B / | A | ; read as matrix B divided by determinant of matrix A.
Practice applying your acquired knowledge about determinant.
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