DETERMINANTS AND CRAMER’S RULE

Mr. Velazquez

Honors Precalculus

DETERMINANT OF A 2X2 MATRIX

−5 24 −2

= −5 −2 − 4 2

= 10− 8 = 2

DETERMINANT OF A 2X2 MATRIX

Evaluate the determinant of each of the following matrices:

(a)

(b)

−2 35 1

−3 24 −1

2-VARIABLE SYSTEMS WITH DETERMINANTS

2-VARIABLE SYSTEMS WITH DETERMINANTS

We will need to calculate three determinants for this technique:

𝐷 =𝑎1 𝑏1𝑎2 𝑏2

𝐷𝑥 =𝑐1 𝑏1𝑐2 𝑏2

𝐷𝑦 =𝑎1 𝑐1𝑎2 𝑐2

ቊ𝑎1𝑥 + 𝑏1𝑦 = 𝑐1𝑎2𝑥 + 𝑏2𝑦 = 𝑐2

The determinant of the coefficients on the left

Same as 𝐷, but with the 𝒙-coefficients replaced with the constants 𝑐1 and 𝑐2

Same as 𝐷, but with the 𝒚-coefficients replaced with the constants 𝑐1 and 𝑐2

𝒙 =𝑫𝒙

𝑫𝒚 =

𝑫𝒚

𝑫

2-VARIABLE SYSTEMS WITH DETERMINANTS

ቊ2𝑥 − 3𝑦 = −11𝑥 + 2𝑦 = 12

Use Cramer’s Rule to solve the following system:

2-VARIABLE SYSTEMS WITH DETERMINANTS

ቊ3𝑥 + 2𝑦 = −12𝑥 − 4𝑦 = 10

Use Cramer’s Rule to solve the following system:

DETERMINANT OF A 3X3 MATRIX

DETERMINANT OF A 3X3 MATRIX

DETERMINANT OF A 3X3 MATRIXAlternatively, copy the first two columns of the matrix in the area to the right. Then draw six diagonals (as shown below) and multiply the numbers in each diagonal. The determinant will be the sum of the “downward” diagonals, minus the sum of the “upward” diagonals.

DETERMINANT OF A 3X3 MATRIXEvaluate the determinant of the following matrix:

−2 1 01 −1 −23 1 0

DETERMINANT OF A 3X3 MATRIXEvaluate the determinant of the following matrix:

−2 1 33 0 1−1 2 3

3-VARIABLE SYSTEMS WITH DETERMINANTS

3-VARIABLE SYSTEMS WITH DETERMINANTS

ቐ

−2𝑥 + 𝑦 = 1𝑥 − 𝑦 − 2𝑧 = 23𝑥 + 𝑦 = 6

Use Cramer’s Rule to solve the following system:

DEPENDENT AND INCONSISTENT SYSTEMS

ቊ2𝑥 − 𝑦 = 6

−6𝑥 + 3𝑦 = 3

𝐷 =2 −1−6 3

= 0

𝐷𝑥 =6 −13 3

= 21

𝐷𝑦 =2 6−6 3

= 42

This system will therefore be inconsistent

CLASSWORK & HOMEWORK

CLASSWORK: CRAMER’S RULE – Use determinants and Cramer’s Rule to solve the following system of equations. (Note: If Cramer’s Rule is not used, you will not receive credit)

HOMEWORK:

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