Determinants, Adjoint and Inverse for CUET Maths

What a determinant is

A fruit seller charges ₹260 for 2 kg of apples and 1 kg of bananas, and ₹230 for 1 kg of apples and 3 kg of bananas. Can the two prices be worked out, and is there only one answer? Before solving anything, one number decides: the determinant of the coefficients. We will come back to the fruit seller at the end of the chapter.

What is the determinant of a matrix?

The determinant is a single number attached to every square matrix, written det A or with vertical bars around the matrix. It decides whether the matrix has an inverse, and unlike a length or an area it can be negative.

For a $1 \times 1$ matrix the determinant is the element itself, so $|[-3]| = -3$: here the bars mean the determinant, not the absolute value.

A $3 \times 3$ determinant is expanded along a row: each element times the $2 \times 2$ determinant left when its row and column are deleted, with the signs $+, -, +$.

Any row or column gives the same value, as long as each element carries its place sign, $(-1)^{i+j}$:

So expand along the row or column with the most zeros. If every element below the leading diagonal is 0, the determinant is just the product of the diagonal elements.

What is the determinant of kA?

For a square matrix of order $n$, $|kA| = k^n|A|$. Multiplying the matrix by $k$ multiplies each of its $n$ rows by $k$, and a factor $k$ comes out of the determinant once for every row, so for a $3 \times 3$ matrix $|2A| = 8|A|$.

Two more facts are used all the time: $|A'| = |A|$, because expanding along a row of $A$ is expanding along a column of $A'$; and a determinant with a row of zeros, or with two equal or proportional rows, is 0.

Area of a triangle

How do you find the area of a triangle using determinants?

Put the three vertices into a determinant with a column of 1s, and take half its absolute value. The absolute value is needed because listing the vertices in another order can flip the sign of the determinant, but never the area.

When one vertex is the origin, this shrinks to $\frac{1}{2}\lvert x_1y_2 - x_2y_1 \rvert$ for the other two.

Three points are collinear exactly when the determinant is 0, because then the "triangle" has no area. That gives the equation of a line too: a point $(x, y)$ is on the line through $(x_1, y_1)$ and $(x_2, y_2)$ when $\begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0$.

Minors and cofactors

What is the difference between a minor and a cofactor?

The minor $M_{ij}$ of an element is the determinant left after deleting its row $i$ and column $j$. The cofactor is the minor with the place sign attached, $A_{ij} = (-1)^{i+j}M_{ij}$, so the two are equal when $i + j$ is even and opposite when it is odd.

With cofactors, the expansion needs no separate signs: a determinant is the sum of the elements of any one row or column, each times its own cofactor.

The adjoint and the inverse

What is the adjoint of a matrix?

The adjoint, $\operatorname{adj} A$, is the transpose of the matrix of cofactors: the cofactor of $a_{ij}$ goes to row $j$, column $i$. For a $2 \times 2$ matrix this comes to a quick rule: swap the two diagonal elements and change the signs of the other two.

The adjoint matters because of one identity, true for every square matrix:

When does a matrix have an inverse?

A square matrix has an inverse exactly when its determinant is not 0. Such a matrix is called non-singular, and a matrix with $|A| = 0$ is singular. Dividing the adjoint identity by $|A|$ gives the inverse:

Taking determinants of the adjoint identity and of $AA^{-1} = I$ gives the facts CUET asks most, for a matrix of order $n$:

Applying the middle one twice, $|\operatorname{adj}(\operatorname{adj} A)| = |A|^{(n-1)^2}$, which is $|A|^4$ for a $3 \times 3$ matrix. And as with the transpose, the inverse of a product reverses the order: $(AB)^{-1} = B^{-1}A^{-1}$.

Consistency of a system of equations

A system of linear equations is consistent if it has at least one solution and inconsistent if it has none. Written as $AX = B$, with the coefficients in $A$, the unknowns in $X$ and the right sides in $B$, the determinant sorts the cases.

When does a system of equations have no solution?

It has no solution when the determinant of its coefficients is 0 and the product of the adjoint with the right-hand column is not the zero matrix. In words, the equations contradict each other: for $2x - y = 3$ and $4x - 2y = 5$, the second left side is twice the first, but 5 is not twice 3.

The matrix method

How do you solve equations by the matrix method?

Write the equations as $AX = B$, check that $|A| \ne 0$, and multiply on the left by the inverse: $X = A^{-1}B$. The order matters, because $BA^{-1}$ is not even defined when $B$ is a column.

Back to the fruit seller. With apples at ₹$a$ a kilogram and bananas at ₹$b$, the prices say $2a + b = 260$ and $a + 3b = 230$. The determinant of the coefficients is $2(3) - 1(1) = 5$, not 0, so there is exactly one answer:

Apples cost ₹110 a kilogram and bananas ₹40.

How CUET asks this chapter

• Values: a $2 \times 2$ or $3 \times 3$ determinant, often with letters or $\sin\theta$ and $\cos\theta$ in it, or the value of $x$ that makes one 0.
• Determinant facts: $|kA|$, $|AB|$, $|\operatorname{adj} A|$, $|A^{-1}|$ and $|\operatorname{adj}(\operatorname{adj} A)|$ from a given $|A|$, often as a match-the-list question.
• Area and collinearity: an area from vertices, or the value of $k$ that gives a set area or makes three points collinear.
• Inverse: a $2 \times 2$ inverse, one element of an adjoint, or the inverse read off a matrix equation such as $A^2 - 3A + I = O$.
• Systems: how many solutions a system has, the $k$ that leaves no unique solution, or one unknown by the matrix method, sometimes in a word problem.

Exam shortcuts

• $|kA| = k^n|A|$, $|\operatorname{adj} A| = |A|^{n-1}$, $|A^{-1}| = \frac{1}{|A|}$.
• A triangular determinant is the product of its diagonal.
• A skew-symmetric matrix of odd order has determinant 0.
• Elements of one row times the cofactors of another row: always 0.
• $2 \times 2$ adjoint: swap the diagonal, change the signs of the other two.
• If $A^2 = kI$, then $A^{-1} = \frac{1}{k}A$.

Common mistakes

• Giving the middle term of a $3 \times 3$ expansion a plus sign.
• Using $k|A|$ for $|kA|$.
• Forgetting the $\frac{1}{2}$ in the area formula, or one of the two values of $k$.
• Taking the cofactor matrix as the adjoint without transposing it.
• Using the adjoint as the inverse when $|A| \ne 1$.
• Deciding "no solution" from $|A| = 0$ alone, without checking $(\operatorname{adj} A)B$.

Quick revision

Calculus starts with how a function changes, and how that change changes: derivatives and second-order derivatives.