How do you find the inverse of a 2x2 matrix?

Formula: A-1 = (1/det) * [[d,-b],[-c,a]] Steps for A = [[a,b],[c,d]]: 1. Calculate det = ad - bc 2. If det = 0, stop — no inverse exists 3. Swap a and d: main diagonal becomes [d, a] 4. Negate b and c: off-diagonal becomes [-b, -c] 5. Divide all four entries by det Example: A = [[1,2],[3,4]] det = 1*4 - 2*3 = 4 - 6 = -2 adj(A) = [[4,-2],[-3,1]] A-1 = (1/-2) * [[4,-2],[-3,1]] = [[-2,1],[1.5,-0.5]] Verify: [[1,2],[3,4]] * [[-2,1],[1.5,-0.5]] = [[-2+3, 1-1],[-6+6, 3-2]] = [[1,0],[0,1]] ✓

What is the adjugate (adjoint) of a 2x2 matrix?

For A = [[a,b],[c,d]], the adjugate is: adj(A) = [[d,-b],[-c,a]] To form the adjugate: - Swap the diagonal entries (a ↔ d) - Negate the off-diagonal entries (b → -b, c → -c) The relationship is: A-1 = adj(A) / det(A) The adjugate is the transpose of the cofactor matrix. For 2x2: - Cofactor of a: +d - Cofactor of b: -c - Cofactor of c: -b - Cofactor of d: +a Transpose these into their positions: [[+d,-b],[-c,+a]] = adj(A)

When does a 2x2 matrix NOT have an inverse?

A 2x2 matrix has no inverse when det(A) = ad - bc = 0. This happens when the rows (or columns) are linearly dependent — one row is a scalar multiple of the other. Examples of singular matrices: - [[2,4],[1,2]]: row 2 = (1/2) * row 1, det = 4-4 = 0 - [[0,0],[1,2]]: row 1 is all zeros, det = 0 - [[3,6],[1,2]]: row 1 = 3 * row 2, det = 6-6 = 0 Geometrically: a singular matrix collapses 2D space onto a line (or a point). This transformation loses information, so it cannot be reversed. In systems of equations Ax = b: - If det = 0 and b is consistent: infinitely many solutions - If det = 0 and b is inconsistent: no solution

How do you verify that a matrix inverse is correct?

Verification: multiply A by A-1 and check the result equals the identity matrix I. For 2x2: A * A-1 = [[1,0],[0,1]] Example: A = [[1,2],[3,4]], A-1 = [[-2,1],[1.5,-0.5]] A * A-1: Row 1: [1*(-2)+2*1.5, 1*1+2*(-0.5)] = [-2+3, 1-1] = [1, 0] ✓ Row 2: [3*(-2)+4*1.5, 3*1+4*(-0.5)] = [-6+6, 3-2] = [0, 1] ✓ Result = [[1,0],[0,1]] = I ✓ Also verify A-1 * A = I (order matters for non-square matrices but for square invertible matrices both should equal I).

What are the properties of matrix inverses?

Key properties of matrix inverses: 1. (A-1)-1 = A The inverse of the inverse is the original matrix. 2. (AB)-1 = B-1 * A-1 The inverse of a product reverses the order (like putting on/taking off socks and shoes). 3. (AT)-1 = (A-1)T Inverse and transpose commute. 4. det(A-1) = 1/det(A) If A scales areas by factor k, A-1 scales by 1/k. 5. (kA)-1 = (1/k) * A-1 Scalar multiples of A. 6. A * A-1 = A-1 * A = I Inverse commutes with the original (for square matrices).

2x2 Matrix Inverse Calculator — Step-by-Step with Verification

The 2x2 Matrix Inverse Calculator finds A-1 = (1/det) x [[d,-b],[-c,a]] with full step-by-step working. Enter any 2x2 matrix and see: the determinant calculation, the adjugate matrix, every inverse entry computed, and a live verification that A x A-1 equals the identity matrix. If det = 0, the calcu...

ENTER MATRIX A

Formula

A⁻¹ = (1/det) × [[d, −b], [−c, a]]

where det = ad − bc

[

]

STEP-BY-STEP

Calculate the determinant

det = ad − bc = (1)(4) − (2)(3) = 4 − 6 = -2

Swap diagonal entries (a ↔ d)

New main diagonal: d=4 (top-left), a=1 (bottom-right)

Negate off-diagonal entries (b, c)

b → −b = -2 (top-right), c → −c = -3 (bottom-left)

Divide every entry by det = -2

a⁻¹ = 4/-2 = -2, b⁻¹ = -2/-2 = 1, c⁻¹ = -3/-2 = 3/2, d⁻¹ = 1/-2 = -1/2

QUICK EXAMPLES

INVERSE MATRIX A⁻¹

1

2

3

4

⁻¹ =

A⁻¹

-2

1

3/2

-1/2

det(A)

-2

1/det

-1/2

det(A⁻¹)

-1/2

A invertible?

Yes ✓

ADJUGATE MATRIX adj(A)

adj(A) = [[d, −b], [−c, a]] = [[4, -2], [-3, 1]]

A⁻¹ = (1/-2) × adj(A)

VERIFICATION: A × A⁻¹ = I

[

1

2

3

4

]

[

-2

1

3/2

-1/2

]

[

1

0

0

1

]

✓

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter the four values of your 2x2 matrix: a (top-left), b (top-right), c (bottom-left), d (bottom-right). The inverse is computed automatically as you type.
2
The step-by-step panel shows all four stages: compute det=ad-bc, swap diagonal (a and d exchange), negate off-diagonal (b and c change sign), then divide every entry by det.
3
The result matrix A-1 is displayed on a dark background. The adjugate matrix adj(A) = [[d,-b],[-c,a]] is shown separately so you can see the intermediate step.
4
The verification panel shows A x A-1 computed explicitly — every entry of the product is calculated and the result should equal the identity matrix [[1,0],[0,1]].
5
If det = 0, the matrix is singular and no inverse exists. The calculator explains why: the rows are linearly dependent, meaning the matrix maps 2D space onto a line and cannot be reversed.

WORKED EXAMPLE

A = [[1,2],[3,4]]. Step 1: det = 1*4 - 2*3 = 4-6 = -2 (not zero, so inverse exists). Step 2: Swap diagonal: a=1 and d=4 exchange -> [[4,2],[3,1]]. Step 3: Negate off-diagonal: b=2->-2, c=3->-3 -> adj(A) = [[4,-2],[-3,1]]. Step 4: Divide by det=-2: A-1 = [[-2,1],[1.5,-0.5]]. Verify: [[1,2],[3,4]] * [[-2,1],[1.5,-0.5]] = [[1,0],[0,1]] ✓.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.