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Determinants - Determinant of a matrix of order one

Grade 12CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A matrix of order 1 is a square matrix that contains only one element, represented as A=[a11]A = [a_{11}].

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The determinant of a matrix of order 1 is defined as the value of the element itself.

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For a 1×11 \times 1 matrix A=[a]A = [a], the determinant is denoted by ∣A∣|A| or det⁡(A)\det(A).

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It is crucial to note that the symbol ∣A∣|A| represents the determinant of the matrix and should not be confused with the absolute value of a number. For example, if the element is negative, the determinant remains negative.

📐Formulae

∣A∣=a11 if A=[a11]|A| = a_{11} \text{ if } A = [a_{11}]

det⁡([a])=a\det([a]) = a

💡Examples

Problem 1:

Find the determinant of the matrix A=[5]A = [5].

Solution:

∣A∣=5|A| = 5

Explanation:

Since the matrix AA is of order 1×11 \times 1, its determinant is equal to the single element contained within the matrix.

Problem 2:

Evaluate ∣B∣|B| if B=[−12]B = [-12].

Solution:

∣B∣=−12|B| = -12

Explanation:

In matrix algebra, the determinant of a 1×11 \times 1 matrix is the value of the element. Unlike the absolute value function, the determinant preserves the sign of the element.

Problem 3:

If C=[sin⁡θ]C = [\sin \theta], find det⁡(C)\det(C).

Solution:

det⁡(C)=sin⁡θ\det(C) = \sin \theta

Explanation:

The rule for order 1 matrices applies to any real or complex value or trigonometric expression; the determinant is simply the entry itself.