May 25

# Learn Properties of Determinants and Operations on Algebraic Expression

A determinant is a special number that may be calculated from a square matrix in linear algebra. Det(P), |P|, or Det P is the determinant of a matrix, say P. Determinants have some useful properties in that they allow us to generate the same results with varied and simpler entry configurations (elements). Reflection property, all-zero property, proportionality or repetition property, switching property, scalar multiple property, sum property, and others are some of the main properties of determinants.

On the other hand, an algebraic expression (or) a variable expression is a set of terms formed by the operations like addition, subtraction, multiplication, division, and so on. To simplify an algebraic expression you must know how to apply operations like Addition and Subtraction of Algebraic Expressions.

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To get more clarity on the use of the aforementioned properties, let’s discuss them in length with examples if necessary to help you solve any determinant or algebraic expression with ease.

## Some Fundamental Properties of Determinants

### Reflection Property

If the determinant’s rows are transformed into columns and the columns into rows, the determinant remains unchanged. This is referred to as the property of reflection.

### All-zero Property

If all of the elements in a row (or column) are zero, the determinant is 0.

### Switching Property

The determinant’s sign is changed by the interchange of any two rows (or columns).

### Scalar Multiple Property

The determinant is multiplied by the same constant if all the elements of its row (or column) are multiplied by a non-zero constant.

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### Sum Property

If a few rows or columns are expressed as a sum of terms, the determinant can be described as a sum of two or more determinants.

∣ a1 a2 a3 ∣    ∣ b1 b2 b3 ∣    ∣ a1+b1 a2+b2 a3+b3 ∣

∣ b1 b2 b3 ∣ + ∣ c1 c2 c3 ∣ = ∣    c1        c2        c3     ∣

∣ c1 c2 c3 ∣    ∣ d1 d2 d3 ∣    ∣    d1        d2        d3     ∣

The first row’s elements represent the sum of words, which may be divided into two distinct determinants. In addition, the new determinants share the same second and third rows as the previous determinant.

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### Triangular Property

Assume that the elements above and below the main diagonal are both equal to zero. In that scenario, the determinant’s value is equal to the product of the diagonal matrix’s members.

∣ a1 a2 a3 ∣     ∣ a1   0   0 ∣

∣  0  b2 b3 ∣ =  ∣ a2 b2   0 ∣ = a1.b2.c3

∣  0   0  c3 ∣      ∣ a3 b3 c3 ∣

### Factor Property

If we set a determinant Δ to zero, it becomes zero.

while assuming the value of x=a,

Then (x-a) is a factor of Δ.

## Basic Operations on Algebraic Expression

The fundamental operations employed in mathematics (particularly in real number systems) are addition, subtraction, multiplication, and so on. These operations are also applicable to algebraic expressions. Let us discuss them in further depth.

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Here is an example of algebraic expressions that can be added:

(1.5ab + 3) + (2.5ab – 2) = (1.5ab + 2.5ab) + (3 + (-2)) = 4ab + 1

### Subtracting Algebraic Expressions

Add the additive inverse of the second expression to the first expression to remove two algebraic expressions. Here is an example of algebraic expressions that can be subtracted:

(3ab + 4) – (2ab – 4) = (3ab + 4) + (-2ab + 4) = (3ab – 2ab) + (4 + 4) = ab + 8

### Multiplying Algebraic Expressions

To multiply two algebraic expressions, multiply each term in the first equation by each term in the second expression and add the results. Here is an example of algebraic expressions being multiplied.

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ab (2ab + 3) = 2a2b2 + 3ab

### Dividing Algebraic Expressions

We factor the numerator and denominator, cancel the possible terms, then simplify the remainder to divide two algebraic expressions. Here is an example of algebraic expressions that have been divided.

(x2 + 5x + 4) / (x + 1) = [ (x + 4) (x + 1) ] / (x + 1) = x + 4